Suppose $F$ is a probability measure supported on $[-L,L]^d$ for some $ L  > 0$ and $\Sigma \in \R^{d \times d}$ is a positive-definite matrix with smallest eigenvalue $\underline{\sigma}^2  > 0 $. Then, for all $\epsilon \in (0,1)$, there exists a discrete probability measure $F'$  with at most  $ D  \max \big \{ ( \log(\epsilon^{-1}))^d, (L / \underline{\sigma})^d ( \log(\epsilon^{-1}))^{d/2}   \big \}  $ support points on $[-L,L]^d$  such that  $\| \varphi_{F, \Sigma} - \varphi_{F',\Sigma}    \|_{L^2} \leq D' \underline{\sigma}^{-d/2} \epsilon  $, where $D,D' > 0$ are universal constants. Furthermore, the support points can be chosen such that $\inf_{i \neq j} \| \mu_i - \mu_j \| \geq \underline{\sigma} \epsilon$.

Let $D$ denote a generic universal constant that may change from line to line. By Lemma <ref>, there exists a discrete measure $F'$ with at most $k^d + 1$ support points on $[-L,L]^d$ such that
\begin{align*}
\left|  \F[F](t) - \F[F'] (t)  \right| \leq  2 \frac{\| t \|^k  (e \sqrt{d} L)^k  }{k^k} \; \; \; \; \forall \; t \in \R^d.
\end{align*}

Observe that $  \left| \varphi_{\Sigma} \right|  = e^{- t' \Sigma t /2}  \leq e^{- \| t \|^2 \underline{\sigma}^2/2}$. From using the preceding bound, we obtain for every $ M > 0$, the estimate
\begin{align*}
\|   \varphi_{F,\Sigma}  - \varphi_{F',\Sigma}   \|_{L^2}^2 & = \int_{\R^d} \left| \varphi_{\Sigma}    (t)        \right|^2   \left|  \F[F](t) - \F[F'](t)    \right|^2 dt \\ & \leq D  \bigg[ \int_{\| t \|> M} \left| \varphi_{\Sigma}    (t)        \right|^2  dt +  \bigg( \frac{L e \sqrt{d}}{k}   \bigg)^{2k}  \int_{\| t \| \leq  M} \| t\|^{2k} dt\bigg] \\ & \leq   D \bigg[    \int_{\| t \|> M} e^{- \| t \|^2 \underline{\sigma}^2 } dt  +  \bigg( \frac{L e \sqrt{d}}{k}   \bigg)^{2k}  \int_{\| t \| \leq  M} \| t\|^{2k} dt\bigg].
\end{align*}

By change of variables (to spherical coordinates) the first integral scales at rate  $D M^{d-2} e^{-M^2 \underline{\sigma}^2 }  \underline{\sigma}^{-2} $  and the second integral scales at rate $D M^{2k+d}/ (2k+d) $. The choice $M = \sqrt{4 \log (\epsilon^{-1})} / \underline{\sigma}   $ leads to
$$ \int_{\| t \|> M} e^{- \| t \|^2 \underline{\sigma}^2 } dt \leq D M^{d-2} e^{-M^2 \underline{\sigma}^2 } \underline{\sigma}^{-2}  \leq D \epsilon^2 \underline{\sigma}^{-d}.     $$

Furthermore, if $k \geq  e^3 \sqrt{d}  \max \{ L M , \log(\epsilon^{-1})  \} $, we have that
\begin{align*}
\bigg( \frac{L e \sqrt{d}}{k}   \bigg)^{2k}  \int_{\| t \| \leq  M} \| t\|^{2k} dt \leq D \bigg( \frac{L e \sqrt{d} M}{k}   \bigg)^{2k} \frac{M^d}{2k+d} \leq D \epsilon^2.
\end{align*}

The claim follows from observing that the number of support points in $F'$ is $ N =    k^d +1  $.

For the final claim regarding the separation of the support points, suppose $F' = \sum_{i=1}^N p_i \delta_{\mu_i}$ is a discrete probability measure that satisfies the requirements of the first part of the Lemma. Let $\mathcal{Z}$ denote a maximal $ \underline{\sigma} \epsilon $ separated subset of $[-L,L]^d$. For each $\mu_i$, select $\mu_i^* \in \arg \min \limits_{t \in \mathcal{Z}} \| \mu_i - t   \| $ and let $F^* = \sum_{i=1}^N p_i \delta_{\mu_i^*} $. From the definition of $\mathcal{Z}$, it follows that $ \sup_{i=1}^N  \| \mu_i  - \mu_i^* \| \leq   \underline{\sigma} \epsilon $. Observe that
\begin{align*}
\|   \varphi_{F',\Sigma} - \varphi_{F^*,\Sigma}  \|_{L^2} & =    \bigg \| \sum_{j=1}^N p_j  \big [ e^{i t \mu_j} -  e^{i t \mu_j^*} \big] e^{-t' \Sigma t /2}             \bigg \|_{L^2} \\ & \leq  D \sum_{j=1}^N p_j \| \big ( e^{i t \mu_j} -  e^{i t \mu_j^*} \big) e^{-t' \Sigma t /2}   \|_{L^2} \\ & \leq D \sup_{j=1,\dots,N } \| \big ( e^{i t \mu_j} -  e^{i t \mu_j^*} \big) e^{-t' \Sigma t /2}   \|_{L^2}.
\end{align*}

Since the mapping $\mu \rightarrow e^{it \mu}$ has Lipschitz constant bounded by $ \| t \|$, we obtain
\begin{align*}
\| \big ( e^{i t \mu_j} -  e^{i t \mu_j^*} \big) e^{-t' \Sigma t /2}   \|_{L^2}^2 &  = \int_{\R^d} \left|  e^{i t \mu_j} -  e^{i t \mu_j^*} \right|^2 e^{- t' \Sigma t} dt \\ & \leq D \| \mu_j - \mu_j^* \|^2 \int_{\R^d}  \| t \|^2 e^{-t' \Sigma t} dt \\ & \leq D \underline{\sigma}^2 \epsilon^2 \int_{\R^d}  \| t \|^2 e^{-t' \Sigma t} dt .
\end{align*}

Since $e^{-t' \Sigma t}  \leq  e^{ - \| t \|^2 \underline{\sigma}^2 }$, the integral on the right scales with rate at most $ D \underline{\sigma}^{-(d+2)}  $. It follows that
\begin{align*}
\| \big ( e^{i t \mu_j} -  e^{i t \mu_j^*} \big) e^{-t' \Sigma t /2}   \|_{L^2} \leq D \underline{\sigma} \epsilon  \underline{\sigma}^{-(d+2)/2} = D  \underline{\sigma}^{-d/2} \epsilon.
\end{align*}

In particular, we obtain
$$ \| \varphi_{F, \Sigma} - \varphi_{F^*,\Sigma}    \|_{L^2} \leq \| \varphi_{F, \Sigma} - \varphi_{F',\Sigma}    \|_{L^2} + \| \varphi_{F^*, \Sigma} - \varphi_{F',\Sigma}    \|_{L^2} \leq D \underline{\sigma}^{-d/2} \epsilon. $$

Finally, note that if $\mu_i^* = \mu_j^*$ for some $i \neq j$, $F^*$ can be reduced to a discrete measure with $N^* \leq N$ unique support points.

Fix any positive definite matrix $\Sigma_0 \in \R^{d \times d}$ and denote by $ \sigma_0^2$, the smallest eigenvalue of $\Sigma_0$. Then, there exists a universal constant $D > 0$ (only depending on $d$) such that for any distribution $P$ and positive definite matrix $\Sigma$ satisfying $\| \Sigma - \Sigma_0 \| \leq \sigma_0^2/2 $, we have
\begin{align*}
\| \varphi_{P,\Sigma} - \varphi_{P,\Sigma_0}  \|_{L^2} \leq D \sigma_0^{-d/2 -1}  \| \Sigma - \Sigma_0 \|.
\end{align*}

For any distribution $P$ and positive definite matrix $\Sigma$, we have that
\begin{align*}
\| \varphi_{P,\Sigma} - \varphi_{P,\Sigma_0}   \|_{L^2}^2  =  \int_{\R^d} \left| \mathcal{F}[P](t)   \right|^2 \left| e^{- t' \Sigma t /2}    -  e^{-t' \Sigma_0 t} \right|^2 dt &  \leq  \int_{\R^d}  \left| e^{- t' \Sigma t /2}    -  e^{- t' \Sigma_0 t/2} \right|^2 dt \\ & = \int_{\R^d}   e^{- t' \Sigma_0 t }   \left| 1 - e^{ t' ( \Sigma_0 - \Sigma)t/2 }   \right|^2 dt.
\end{align*}

The mapping $t \rightarrow  e^{ t' ( \Sigma_0 - \Sigma)t/2 }$ has gradient norm at most $ e^{\| t \|^2 \| \Sigma - \Sigma_0  \|/2  } \| \Sigma - \Sigma_0 \| \| t \| $. If the bound $  \| \Sigma - \Sigma_0  \| \leq \sigma_0^2/2 $ holds, it follows that
\begin{align*}
\int_{\R^d}   e^{- t' \Sigma_0 t }   \left| 1 - e^{ t' ( \Sigma_0 - \Sigma)t/2 }   \right|^2 dt & \leq  \int_{\R^d}   e^{- \| t\|^2 \sigma_0^2 }   \left| 1 - e^{ t' ( \Sigma_0 - \Sigma)t/2 }   \right|^2 dt \\ & \leq \| \Sigma - \Sigma_0  \|^2   \int_{\R^d}   e^{- \| t\|^2 \sigma_0^2/2 } \| t \|^2   dt .
\end{align*}

The claim follows from observing that the integral on the right scales with rate at most $D \sigma_0^{-(d+2)}$.

Suppose $(W ,Z) \in \R^{d^*} \times \R^{d} $ for some $d^*,d \in \mathbb{N}$  and $\E\big( \| Z \|^2  \big) < \infty , \E\big( \| W \|^2  \big) < \infty $. For every $t \in \R^d$, define
\begin{align*}
\chi_t(W,Z)  =  We^{\mathbf{i} t' Z} = W\cos(t'Z) + \mathbf{i}  W \sin(t'Z).
\end{align*}

Then, there exists a universal constant $D > 0$ such that for every $T > 0$ we have

\begin{align*}
\E  \bigg( \sup_{ \| t \|_{\infty} \leq T} \| \E_n \big[ \chi_t(Z)     \big] - \E \big[ \chi_t(Z)     \big] \| \bigg) \leq D \frac{ \max \big \{ \sqrt{\log T} , 1  \big \}  }{\sqrt{n}}.
\end{align*}

It suffices to show the result for the real and imaginary part separately. We verify it for the real part, the imaginary part is completely analogous. If $W= (W_1,W_2,\dots,W_{d_1})$, it suffices to verify the result for each vector sub component $ W_k e^{\mathbf{i} t' Z} $ where $k \in \{1 , \dots, d^* \}$. Fix any $k \in \{1 , \dots , d^* \}$. For the remainder of this proof, we continue assuming $\chi_t(W,Z) =  W_k \cos(t'Z)$. Define the class of functions $\mathcal{F} = \{   \chi_t(W,Z) : \| t \|_{\infty} \leq T \}$. Since $\|W \|$ is an envelope of $\mathcal{F}$, an application of <cit.> implies that  there exists a universal constant $ L > 0 $ such that
\begin{align*}
\E  \bigg(\sup_{ \| t \|_{\infty} \leq T}  \left| \E_n \big[ \chi_t(W,Z)     \big] - \E \big[ \chi_t(W,Z)     \big] \right| \bigg) \leq \frac{L}{\sqrt{n}} \int_{0}^{8 \| W \|_{L^2(\mathbb{P})}     }  \sqrt{\log N_{[]}(\mathcal{F}, \| . \|_{L^2(\mathbb{P})},  \epsilon  ) } d \epsilon.
\end{align*}

Let $\{  t_i \}_{i=1}^M  $  denote a minimal $\delta > 0$ covering of  $[-T,T]^{d}$. Define the functions
$$ e_{i}(W,Z) = \sup_{ t \in \R^d :   \| t \|_{\infty} \leq T , \|  t- t_i \|_{\infty} < \delta  } \ \big| \:   \chi_t(W,Z)  -   \chi_{t_i}(W,Z)   \:   \big| \; \; \; \; \; i=1,\dots,M . $$

It follows that $ \big \{   \chi_{t_i}(W,Z)   - e_i \; ,  \;  \chi_{t_i}(W,Z)  + e_i   \big \}_{i=1}^M  $ is a bracket covering for $\mathcal{F}$. Since the mapping $t \rightarrow \chi_t(W,Z)$ has Lipschitz constant bounded by $ \| W \| \|Z \| $, Cauchy-Schwarz implies that $
\| e_i \|_{L^2(\mathbb{P})}^2    \leq  \| W \|_{L^2(\mathbb{P})}^2 \| Z \|_{L^2(\mathbb{P})}^2  \delta^2$. Since $M \leq (3 T \delta^{-1})^d$, it follows that there exists a universal constant $L >0$ such that
\begin{align*}
\int_{0}^{8 \| W \|_{L^2(\mathbb{P})}     }  \sqrt{\log N_{[]}(\mathcal{F}, \| . \|_{L^2(\mathbb{P})},  \epsilon  ) } d \epsilon \leq L \max \big \{ \sqrt{\log T} ,1   \big  \}.
\end{align*}

Suppose $(W ,Z) \in \R^{d^*} \times \R^{d} $ for some $d^*,d \in \mathbb{N}$  and $\E\big( \| Z \|^2  \big) < \infty , \E\big( \| W \|^2  \big) < \infty $. For every $t \in \R^d$, define
\begin{align*}
\chi_t(W,Z)  =  We^{\mathbf{i} t' Z} = W\cos(t'Z) + \mathbf{i}  W \sin(t'Z).
\end{align*}

Then, there exists a universal constant $L > 0$ such that for any sequence $T_n \uparrow \infty$ with $\log(T_n) = o(n)$, we have that

\begin{align*}
\mathbb{P}  \bigg( \sup_{ \| t \|_{\infty} \leq T_n} \left| \E_n \big[ \chi_t(W,Z)     \big] - \E \big[ \chi_t(W,Z)     \big] \right| \leq D \frac{\sqrt{\log T_n}}{\sqrt{n}} \bigg)  \rightarrow 1.
\end{align*}

It suffices to show the result for the real and imaginary part separately. We verify it for the real part, the imaginary part is completely analogous. If $W= (W_1,W_2,\dots,W_{d^*})$, it suffices to verify the result for each vector sub component $ W_k e^{\mathbf{i} t' Z} $ where $k \in \{1 , \dots, d^* \}$. Fix any $k \in \{1 , \dots , d^* \}$. For the remainder of this proof, we continue assuming $\chi_t(W,Z) =  W_k \cos(t'Z)$. For a given sequence of deterministic constants $L_n \uparrow \infty$, define
\begin{align*}  &  \chi_{1,t}(W,Z)  =  \chi_t(W,Z)  \mathbbm{1} \big \{  \| W \| \leq L_n   \big \} \;, \\ &  \chi_{2,t}(W,Z)  =  \chi_t(W,Z)  \mathbbm{1} \big \{  \| W \| > L_n   \big \}  \; .      \end{align*}

Observe that
\begin{align*}
(\E_n - \E_{})[\chi_t(W,Z) ]   =\sum_{i=1}^n  \Xi_{1,t} (W_i,Z_i)  + \sum_{i=1}^n \Xi_{2,t} (W_i,Z_i) \; ,
\end{align*}

where $ \Xi_{1,t} (W,Z) = n^{-1} [\chi_{1,t}(W,Z) - \E_{} \chi_{1,t}(W,Z) ] $ and $ \Xi_{2,t} (W,Z) = n^{-1} [\chi_{2,t}(W,Z) - \E_{} \chi_{2,t}(W,Z) ]$.

First, we derive a bound for $ \sum_{i=1}^n \Xi_{2,t} (W_i,Z_i)$. Observe that
\begin{align*}
\mathbb{P} \bigg( \sup_{\| t \|_{\infty} \leq T_n}  \left|  \sum_{i=1}^n \Xi_{2,t} (W_i,Z_i)    \right| > \frac{\sqrt{\log(T_n)}}{\sqrt{n}}   \bigg)  & \leq  \frac{\sqrt{n}}{ \sqrt{\log(T_n)} } \E \bigg( \sup_{\| t \|_{\infty} \leq T_n} \sum_{i=1}^n \left|  \Xi_{2,t} (W_i,Z_i)   \right|      \bigg) \\ & \leq  \frac{2 \sqrt{n}}{ \sqrt{\log(T_n)} } \E \big( \| W \|  \mathbbm{1} \big \{ \|W \| > L_n    \big \}   \big) \\ & \leq \frac{2 \sqrt{n}}{ \sqrt{\log T_n} L_n }  \E \big( \| W \|^2  \mathbbm{1} \big \{ \|W \| > L_n    \big \}   \big) .
\end{align*}

Since $ \E\big( \| W \|^2  \big) < \infty $, the term on the right is $o(1)$ when $L_n = \sqrt{n}/ \sqrt{\log T_n}$.

It remains to bound the first sum $\sum_{i=1}^n  \Xi_{1,t} (W_i,Z_i) $ when $L_n = \sqrt{n}/ \sqrt{\log T_n}$. Observe that
\begin{align*}
&  \sup_{\| t \|_{\infty} \leq T_n} \E \big[ \left| \Xi_{1,t} (W,Z)  \right|^2    \big] \leq n^{-2} \| W \|_{L^2(\mathbb{P})}^2 \\ &  \sup_{\| t \|_{\infty} \leq T_n} \left| \Xi_{1,t} (W,Z)     \right| \leq 2 n^{-1} L_n.
\end{align*}

The preceding bounds, Lemma <ref> and   <cit.> imply that there exists universal constants $C,D > 0$ which satisfy
\begin{align*}
& \mathbb{P}  \bigg( \sup_{  \|t  \|_{\infty} \leq T_n} \left| \E_n \big[ \chi_t(Z)     \big] - \E \big[ \chi_t(Z)     \big] \right| >   D \frac{\sqrt{\log T_n}}{\sqrt{n}}  + x  \bigg) \\ & \leq  \exp \bigg(  - \frac{x^2}{C n^{-1} \big[ L_n \sqrt{\log T_n}/ \sqrt{n} + \| W \|_{ L^2(\mathbb{P})   }^2 + x L_n   ]  }     \bigg)
\end{align*}

for all $x > 0$. The choice $x =  \sqrt{\log T_n} / \sqrt{n}  $  with  $L_n = \sqrt{n} / \sqrt{\log T_n}$ yields

\begin{align*}
\mathbb{P}  \bigg( \sup_{  \|t  \|_{\infty} \leq T_n} \left| \E_n \big[ \chi_t(Z)     \big] - \E \big[ \chi_t(Z)     \big] \right| >   D \frac{\sqrt{\log T_n}}{\sqrt{n}}  +  \frac{\sqrt{\log T_n}}{\sqrt{n}}   \bigg) \leq \exp \big(  - E \log(T_n)       \big)
\end{align*}

for some universal constant $ E > 0$. Since $T_n \uparrow \infty$, the claim follows.

Suppose $F$ is a probability measure supported on $[-L,L]^d$ for some $ L  > 0$ and $\Sigma \in \R^{d \times d}$ is a positive-definite matrix. Then, for all $\epsilon \in (0,1)$, there exists a discrete probability measure $F'$  with at most  $ D  \max \big \{ ( \log(\epsilon^{-1}))^d, L^d T^d  \big \}  $ support points on $[-L,L]^d$  such that  $\| \varphi_{F, \Sigma} - \varphi_{F',\Sigma}    \|_{ \mathbb{B}(T) } \leq D'  \epsilon  $, where $D,D' > 0$ are universal constants. Furthermore, the support points can be chosen such that $\inf_{i \neq j} \| \mu_i - \mu_j \| \geq T^{-(d+2)/2} \epsilon $.

Let $D$ denote a generic universal constant that may change from line to line. By Lemma <ref>, there exists a discrete measure $F'$ with at most $k^d + 1$ support points on $[-L,L]^d$ such that
\begin{align*}
\left|  \F[F](t) - \F[F'] (t)  \right| \leq  2 \frac{\| t \|^k  (e \sqrt{d} L)^k  }{k^k} \; \; \; \; \forall \; t \in \R^d.
\end{align*}

From using the preceding bound and noting that the eigenvalues of $\Sigma$ are non-negative, we obtain for every $ M > 0$, the estimate

\begin{align*}
\|   \varphi_{F,\Sigma}  - \varphi_{F',\Sigma}   \|_{ \mathbb{B}(T) }^2 & \leq \int_{\| t \|_{\infty} \leq T} \left| \varphi_{\Sigma}    (t)        \right|^2   \left|  \F[F](t) - \F[F'](t)    \right|^2 dt  \\ & \leq 2  \bigg( \frac{L e \sqrt{d}}{k} \bigg)^{2k}  \int_{\| t \|_{\infty} \leq T}  e^{- t' \Sigma t  } \| t \|^{2k} dt \\ & \leq 2  \bigg( \frac{L e \sqrt{d}}{k} \bigg)^{2k}  \int_{\| t \|_{\infty} \leq T} \| t \|^{2k} dt \\ & \leq D  \bigg( \frac{L e \sqrt{d} T }{k} \bigg)^{2k}  T^d.
\end{align*}

The quantity on the right is bounded above by $D \epsilon^2 $ if  $k \geq  e^3 \sqrt{d}  \max \{ L T , \log(\epsilon^{-1})  \} $. The claim follows from observing that the number of support points in $F'$ is $ N =    k^d +1  $.

For the final claim regarding the separation of the support points, suppose $F' = \sum_{i=1}^N p_i \delta_{\mu_i}$ is a discrete probability measure that satisfies the requirements of the first part of the Lemma. Let $\mathcal{Z}$ denote a maximal $ T^{-(d+2)/2} \epsilon $ separated subset of $[-L,L]^d$. For each $\mu_i$, select $\mu_i^* \in \arg \min \limits_{t \in \mathcal{Z}} \| \mu_i - t   \| $ and let $F^* = \sum_{i=1}^N p_i \delta_{\mu_i^*} $. From the definition of $\mathcal{Z}$, it follows that $ \sup_{i=1}^N  \| \mu_i  - \mu_i^* \| \leq   \underline{\sigma} \epsilon $. Observe that
\begin{align*}
\|   \varphi_{F',\Sigma} - \varphi_{F^*,\Sigma}  \|_{ \mathbb{B}(T)} & =    \bigg \| \sum_{j=1}^N p_j  \big [ e^{\mathbf{i} t \mu_j} -  e^{\mathbf{i} t \mu_j^*} \big] e^{-t' \Sigma t /2}             \bigg \|_{ \mathbb{B}(T)} \\ & \leq  D \sum_{j=1}^N p_j \| \big ( e^{\mathbf{i} t \mu_j} -  e^{\mathbf{i} t \mu_j^*} \big) e^{-t' \Sigma t /2}   \|_{ \mathbb{B}(T)} \\ & \leq D \sup_{j=1,\dots,N } \| \big ( e^{\mathbf{i} t \mu_j} -  e^{\mathbf{i} t \mu_j^*} \big) e^{-t' \Sigma t /2}   \|_{ \mathbb{B}(T)}.
\end{align*}

Since the mapping $\mu \rightarrow e^{\mathbf{i}t \mu}$ has Lipschitz constant bounded by $ \| t \|$, we obtain
\begin{align*}
\| \big ( e^{\mathbf{i} t \mu_j} -  e^{\mathbf{i} t \mu_j^*} \big) e^{-t' \Sigma t /2}   \|_{ \mathbb{B}(T)}^2 &  \leq \int_{\|t \|_{\infty} \leq T} \left|  e^{\mathbf{i} t \mu_j} -  e^{\mathbf{i}t \mu_j^*} \right|^2 e^{- t' \Sigma t} dt \\ & \leq D \| \mu_j - \mu_j^* \|^2 \int_{\|t \|_{\infty} \leq T}  \| t \|^2 e^{-t' \Sigma t} dt \\ & \leq D   T^{-(d+2)} \epsilon^2 \int_{\|t \|_{\infty} \leq T}  \| t \|^2  dt.
\end{align*}

The  integral on the right scales with rate at most $ D  T^{d+2}  $. It follows that
$$ \| \varphi_{F, \Sigma} - \varphi_{F^*,\Sigma}    \|_{\mathbb{B}(T)} \leq \| \varphi_{F, \Sigma} - \varphi_{F',\Sigma}    \|_{\mathbb{B}(T)} + \| \varphi_{F^*, \Sigma} - \varphi_{F',\Sigma}    \|_{\mathbb{B}(T)} \leq D \epsilon. $$

Finally, note that if $\mu_i^* = \mu_j^*$ for some $i \neq j$, $F^*$ can be reduced to a discrete measure with $N^* \leq N$ unique support points.

Fix any positive definite matrix $\Sigma_0 \in \R^{d \times d}$ and denote by $ \sigma_0^2$, the smallest eigenvalue of $\Sigma_0$. Then, there exists a universal constant $D > 0$ (only depending on $d$) such that for any distribution $P$ and positive definite matrix $\Sigma$ satisfying $\| \Sigma - \Sigma_0 \| \leq \sigma_0^2/2 $, we have
\begin{align*}
\| \varphi_{P,\Sigma} - \varphi_{P,\Sigma_0}  \|_{ \mathbb{B}(T) } \leq D T^{(d+2)/2}   \| \Sigma - \Sigma_0 \|.
\end{align*}

For any distribution $P$ and positive definite matrix $\Sigma$, we have that
\begin{align*}
\| \varphi_{P,\Sigma} - \varphi_{P,\Sigma_0}   \|_{\mathbb{B}(T_n)}^2  \leq  \int_{\|t \|_{\infty} \leq T} \left| \mathcal{F}[P](t)   \right|^2 \left| e^{- t' \Sigma t /2}    -  e^{-t' \Sigma_0 t} \right|^2 dt &  \leq  \int_{\|t \|_{\infty} \leq T}  \left| e^{- t' \Sigma t /2}    -  e^{- t' \Sigma_0 t/2} \right|^2 dt \\ & = \int_{\|t \|_{\infty} \leq T}   e^{- t' \Sigma_0 t }   \left| 1 - e^{ t' ( \Sigma_0 - \Sigma)t/2 }   \right|^2 dt.
\end{align*}

The mapping $t \rightarrow  e^{ t' ( \Sigma_0 - \Sigma)t/2 }$ has gradient norm at most $ e^{\| t \|^2 \| \Sigma - \Sigma_0  \|/2  } \| \Sigma - \Sigma_0 \| \| t \| $. If the bound $  \| \Sigma - \Sigma_0  \| \leq \sigma_0^2/2 $ holds, it follows that
\begin{align*}
\int_{\|t \|_{\infty} \leq T}   e^{- t' \Sigma_0 t }   \left| 1 - e^{ t' ( \Sigma_0 - \Sigma)t/2 }   \right|^2 dt & \leq  \int_{\|t \|_{\infty} \leq T}   e^{- \| t\|^2 \sigma_0^2 }   \left| 1 - e^{ t' ( \Sigma_0 - \Sigma)t/2 }   \right|^2 dt \\ & \leq \| \Sigma - \Sigma_0  \|^2   \int_{\|t \|_{\infty} \leq T}   e^{- \| t\|^2 \sigma_0^2/2 } \| t \|^2   dt \\ & \leq  D \| \Sigma - \Sigma_0  \|^2 T^{d+2}.
\end{align*}

Suppose $F \sim \text{DP}_{\alpha} $ where the base measure $\alpha$ is a Gaussian measure on $\R^d$. Fix any $q \in \mathbb{N}$. Then, there exists a universal constant $C > 0$ such that for any sequence $u_n \uparrow \infty$,
\begin{align*}
\mathbb{P} \bigg(  \bigg \{  \int \|x\|^q dF(x) \bigg \}^{1/q} > u_n    \bigg) \leq 2 e^{- C u_n^2}
\end{align*}

holds for all sufficiently large $n$.

Suppose $\alpha = N(\mu,\Sigma)$ for some mean vector $\mu \in \R^d$ and positive definite covariance matrix $\Sigma \in \mathbf{S}_+^d$. Without loss of generality, it suffices to verify the result with  $\mu=0$. By the stick breaking representation of $\text{DP}_{\alpha}$ we can write
$$  F \stackrel{d}{=} \sum_{i=1}^{\infty} p_i \delta_{Z_i} \; \; , \; \; \int \|x\|^q dF(x) \stackrel{d}{=} \sum_{i=1}^{\infty} p_i \| Z_i \|^q, $$

where $(Z_1,Z_2,\dots) \stackrel{i.i.d}{\sim} \alpha $ is independent of $(p_1,p_2,\dots)$ and $(p_i)_{i=1}^{\infty}$ are non-negative random variables with $\sum_{i=1}^{\infty} p_i =1$. For an element $w = (w_1,w_2,\dots)$ with $w_i \in \R^d$, the $\ell^q(\R^d)$ norm is given by $ \| w \|^q = \sum_{i=1}^{\infty} \| w _i \|^q   $. Conditional on $(p_i)_{i=1}^{\infty}$, let $ \Z = ( p_i^{1/q} Z_1  , p_2^{1/q} Z_2  , \dots )$. Define $\gamma = (\E \| Z_1 \|^q )^{1/q}$ and note that $ \E \| \Z \|^q = \sum_{i=1}^{\infty} p_i \E \| Z_i \|^q   = \gamma^q  < \infty$. In particular, we can view $\Z$ (defined conditional on $(p_i)_{i=1}^{\infty}$) as a mean-zero Gaussian random element on $\ell^q(\R^d)$. From an application of <cit.>, it follows that
\begin{align*}
\mathbb{P} \bigg(  \| \Z \| > u + \gamma     \bigg) \leq 2 e^{- \frac{u^2}{2 \gamma^2}} \; \; \; \; \forall \; u > 0.
\end{align*}

The bound holds conditionally. However, as the term on the right is independent of $(p_i)_{i=1}^{\infty}$, it also holds unconditionally and we obtain
\begin{align*}
\mathbb{P} \bigg( \bigg \{  \int \|x\|^q dF(x) \bigg \}^{1/q} > u + \gamma  \bigg) \leq 2 e^{- \frac{u^2}{2 \gamma^2}} \; \; \; \; \forall \; u > 0.
\end{align*}

The claim follows by letting $u = u_n$ and noting that $u_n > 2 \gamma$ for all sufficiently large $n$.

For $d \geq 2$, let $\Omega \subset \R^d $ denote a ball (with respect to any norm) of radius $R$. Let $f: \Omega \rightarrow \mathbb{C}$ be such that $\nabla f \in C(\overline{\Omega})$. Then, there exists a universal constant $C > 0$ such that
\begin{align*}
\| f \|_{L^2(\Omega)} \leq C \bigg( R \| \nabla f  \|_{L^2(\Omega)} + \sqrt{R} \| f \|_{L^2(\partial \Omega)}    \bigg) \; ,
\end{align*}

where $ L^2(\partial \Omega)$ is the $L^2$ norm on $\partial \Omega$ with respect to the $d-1$ dimensional Hausdorff measure.

Let $\lambda(.)$ denote the Lebesgue measure on $\R^d$. Denote the $d-1$ dimensional Hausdorff measure by$\mathcal{H}^{d-1}$. Define the exponents
$$ p = \frac{2d}{d+2} \; , \; q =  \frac{2(d-1)}{d} . $$

An application of <cit.> on  the real and complex parts of $f$ separately implies that there exists a universal constant $D > 0$ (that depends only on $p$) such that
\begin{align*}
\| f \|_{L^2(\Omega)} \leq D \big(  \|  \nabla f \|_{L^p( \Omega )} + \| f \|_{L^q(\partial \Omega)}    \big).
\end{align*}

Since $\Omega$ is a ball of radius $R$, there exists a universal constant $C > 0$ such that
$$ \mathcal{H}^{d-1}(\partial \Omega) \leq C R^{d-1} \;  , \;  \lambda(\Omega) \leq C R^d.   $$

By Hölder's inequality, it follows that
\begin{align*}
\|  \nabla f \|_{L^p(\Omega)}^p = \int_{\Omega}  \|  \nabla f  \|^p d \lambda = \int_{\Omega}  \|  \nabla f  \|^{\frac{2d}{d+2}} d \lambda \leq \bigg( \int_{\Omega}  \|  \nabla f  \|^2  d \lambda \bigg)^{\frac{d}{d+2}}  \big[ \lambda(\Omega) \big]^{\frac{2}{d+2}}.
\end{align*}

Taking the $p = 2d/(d+2)$ root on both sides of the preceding inequality and using the estimate on $\lambda(\Omega)$ implies that there exists a universal constant $C > 0$ such that
\begin{align*}
\|  \nabla f \|_{L^p(\Omega)} \leq  C \|  \nabla f  \|_{L^2(\Omega)} R.
\end{align*}

Similarly, for the boundary term, Hölder's inequality yields
\begin{align*}
\|   f \|_{L^q( \partial \Omega)}^q = \int_{\partial \Omega} \left| f \right|^q d \mathcal{H}^{d-1} =  \int_{\partial \Omega} \left| f \right|^{\frac{2(d-1)}{d}} d \mathcal{H}^{d-1} \leq \bigg(  \int_{\partial \Omega} \left| f \right|^2 d \mathcal{H}^{d-1}   \bigg)^{\frac{d-1}{d}} \big[ \mathcal{H}^{d-1} ( \partial \Omega)   ]^{1/d}.
\end{align*}

Taking the $q = 2(d-1)/d$ root on both sides and using the estimate on $\mathcal{H}^{d-1}(\partial \Omega)$ implies that there exists a universal constant $C > 0$ such that
\begin{align*}
\|   f \|_{L^q( \partial \Omega)} \leq C  \|  f  \|_{L^2(\partial \Omega)} \sqrt{R}.
\end{align*}

The proof proceeds through several steps which we outline below. We use $D > 0$ as a generic universal constant that may change from line to line. For ease of notation, we suppress the dependence of $m = m_n$ and $G = G_n$ on $n$

Depending on whether the model is mildly or severely ill-posed, define $\lambda$ as follows.

\begin{align*}
\lambda = \begin{cases}
\max \{ \chi^{-1}(d+2) + d/2 , d+1    \} & \text{mildly ill-posed} \\ \max \{ \chi^{-1}(d+2) + d/2 , d+1  , d/ \zeta  + 1/2  \} & \text{severely ill-posed}.
\end{cases}
\end{align*}

Let $\epsilon_n^2 = n^{-1} (\log n)^{\lambda}  $.

First, we derive a lower bound for the normalizing constant of the posterior measure.  Specifically, we aim to show that there exists a $C > 0 $ such that
\begin{align}
\label{t1lb} \int  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T_n)  }^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) \geq \exp \big(   - C n    \epsilon_{n}^2   \big)
\end{align}

holds with $\mathbb{P}$ probability approaching $1$.

Since $m = m_n \asymp n$, an application of Lemma <ref> implies that
\begin{align*}
&  \| \widehat{\varphi}_{Y}  - \varphi_{Y}  \|_{\mathbb{B}(T_n)}^2  =   \int_{\mathbb{B}(T_n)}   \left|  \widehat{\varphi}_Y(t) - \varphi_Y(t)    \right|^2 dt \leq D  \frac{ T_n^d  \log(T_n)  }{n}  \; ,  \\ & \| \widehat{\varphi}_{\epsilon}  - \varphi_{\epsilon}  \|_{\mathbb{B}(T_n)}^2 = \int_{\mathbb{B}(T_n)}   \left|  \widehat{\varphi}_\epsilon(t) - \varphi_\epsilon(t)    \right|^2 dt \leq D \frac{ T_n^d \log(T_n)  }{n} .
\end{align*}

with $\mathbb{P}$ probability approaching $1$. On this set (and using that $ \left| \varphi_{P,\Sigma} \right|  \leq 1$ ) we obtain
\begin{align*}
\| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2  & \leq D \bigg( \| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2    +  \frac{ T_n^d  \log (T_n)}{n}     \bigg) \\ & \leq D \bigg(   \| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2  + \epsilon_n^2  \bigg) \; ,
\end{align*}

where the last inequality follows from the definition of $T_n$ and $\epsilon_n^2$.

It follows that

\begin{align*}
& \int  \exp \bigg( - nD \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma)  \\ & \geq \exp(-n D \epsilon_n^2 )   \int  \exp \bigg( - nD \| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) .
\end{align*}

By Condition <ref>, the mixing distribution $F_0$ satisfies $F_0( t \in \R^d :  \| t \|  > z) \leq C \exp(- C' z^{\chi})$. There exists a universal constant $R > 0$ such that the cube $I_n =[-R  (\log \epsilon_n^{-1})^{1/\chi},R  (\log \epsilon_n^{-1})^{1/\chi} ]^d $ satisfies $ 1- F_0(I_n) \leq D \epsilon_n$. Denote the probability measure induced from the restriction of $F_0$ to $I_n$ by
$$ \overline{F}_0(A) = \frac{F(A \cap I_n)}{F(I_n)}  \; \; \; \; \; \forall \; \; \text{Borel}  \; A \subseteq \R^d. $$

Note that the restricted probability measure satisfies
\begin{align*}
\sup_{t \in \R^d} \left| \varphi_{F_0} (t) -  \varphi_{\overline{F}_0}(t) \right| =  \sup_{t \in \R^d} \left|  \int_{\R^d} e^{\mathbf{i} t' x} d(F - \overline{F}_0 ) (x)     \right|   \leq \| F_0 - \overline{F}_0   \|_{TV}  \leq 1- F_0(I_n) \leq D \epsilon_n.
\end{align*}

By Condition <ref>, $f_X = \phi_{\Sigma_0} \star F_0$. As the eigenvalues of $\Sigma_0$ are bounded away from zero, it follows that

\begin{align*}
\| \varphi_Y - \varphi_{\epsilon} \varphi_{ \overline{F}_0, \Sigma_0  }   \|_{\mathbb{B}(T_n)}^2   \leq D   \| \varphi_Y - \varphi_{\epsilon} \varphi_{ \overline{F}_0, \Sigma_0  }   \|_{L^2}^2  & =  \| \varphi_{\epsilon} \varphi_{F_0, \Sigma_0} - \varphi_{\epsilon} \varphi_{ \overline{F}_0, \Sigma_0  }   \|_{L^2}^2  \\ &      = \int_{\R^d} \left|  \varphi_{\Sigma_0} (t)  \right|^2 \left|  \varphi_{\epsilon} (t)  \right|^2 \left|  \varphi_{F_0}(t)  - \varphi_{\overline{F}_0}(t) \right|^2 dt \\ & \leq  \int_{\R^d} \left|  \varphi_{\Sigma_0} (t)  \right|^2 \left|  \varphi_{F_0}(t)  - \varphi_{\overline{F}_0}(t) \right|^2 dt      \\    &  \leq  D \epsilon_n^2 \int_{\R^d} e^{- t' \Sigma_0 t} dt \\ & \leq D \epsilon_n^2.
\end{align*}

Let $\iota = \max \{ d, d/\chi + d/2   \} $. By Lemma <ref>, there exists a discrete probability measure $ F_0^* = \sum_{i=1}^N p_i \delta_{\mu_i}  $ where $N = D  \big(\log( \epsilon_n^{-1}) \big)^{ \iota  }$ and $\mu_i \in I_{n}$,  that satisfies

$$ \|  \varphi_{\epsilon} \varphi_{ \overline{F}_0, \Sigma_0  }  - \varphi_{\epsilon} \varphi_{ F_0^* , \Sigma_0  }       \|_{\mathbb{B}(T_n)} \leq D \|  \varphi_{ \overline{F}_0, \Sigma_0  }  - \varphi_{ F_0^* , \Sigma_0  }       \|_{L^2} \leq D \epsilon_n.$$

From the second claim of Lemma <ref>, we can also assume without loss of generality that the support points separation satisfies $\inf_{k \neq j} \| \mu_k  - \mu_j \| \geq c_0 \epsilon_n$ for some constant $c_0 > 0$ (depending only on $\Sigma_0$). From combining the preceding bounds, it follows that
\begin{align*}
& \int  \exp \bigg( - nD \| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \exp(-n D \epsilon_n^2 ) \int  \exp \bigg( - nD \| \varphi_{\epsilon} \varphi_{ \overline{F}_0, \Sigma_0  }      - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ & \geq  \exp(-n D \epsilon_n^2 ) \int  \exp \bigg( - nD \| \varphi_{\epsilon} \varphi_{ F_0^* , \Sigma_0  }      - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma).
\end{align*}

Observe that if $\| \Sigma - \Sigma_0 \| $ is sufficiently small, Lemma <ref> implies that for any distribution $P$ we have
\begin{align*} \| \varphi_{\epsilon}  \varphi_{P,\Sigma} - \varphi_{\epsilon}  \varphi_{P,\Sigma_0}     \|_{\mathbb{B}(T_n)}^2  \leq
D \| \varphi_{P,\Sigma} - \varphi_{P,\Sigma_0}   \|_{L^2}^2 \leq D \| \Sigma - \Sigma_0 \|^2.
\end{align*}

In particular, for all such $(P,\Sigma)$, this implies
$$ \|  \varphi_{\epsilon} \varphi_{ F_0^* , \Sigma_0  }    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}     \|_{\mathbb{B}(T_n)} \leq  D \|  \varphi_{\epsilon} \varphi_{ F_0^* , \Sigma_0  }  - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma_0}     \|_{L^2}  + D \| \Sigma - \Sigma_0 \|  .  $$

From the preceding bounds, it follows that

\begin{align*}
&  \int  \exp \bigg( - nD \| \varphi_{\epsilon} \varphi_{ F_0^* , \Sigma_0  }      - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \int_{(P,\Sigma) : \| \Sigma - \Sigma_0  \| \leq D \epsilon_n , \|   \varphi_{ F_0^* , \Sigma_0  }  -  \varphi_{P,\Sigma_0}     \|_{L^2} \leq  D \epsilon_n   }  \exp \bigg( - nD \| \varphi_{\epsilon} \varphi_{ F_0^* , \Sigma_0  }      - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \exp (-n D \epsilon_n^2) \int_{(P,\Sigma) : \| \Sigma - \Sigma_0  \| \leq  D \epsilon_n , \|   \varphi_{ F_0^* , \Sigma_0  }  -  \varphi_{P,\Sigma_0}     \|_{L^2} \leq D \epsilon_n   }   d \nu_{\alpha,G} (P,\Sigma)   .
\end{align*}

Let $V_i = \{ t \in \R^d : \| t - \mu_i \| \leq  c_0 \epsilon_n / 2  \} $ for $i=1,\dots,N$ and $V_0 = \R^d \setminus \bigcup_{i=1}^N V_i $. From the definition of the $\{\mu_i \}_{i=1}^N$, it follows that $\{V_0, V_1, \dots , V_N  \}$ is a disjoint partition of $\R^d$. For any fixed $(P,\Sigma)$, an application of  Lemma <ref> yields
\begin{align*}
\| \varphi_{P, \Sigma_0} - \varphi_{ F_0^* , \Sigma_0}  \|_{L^2} & \leq D \bigg( \epsilon_n^{d}  +  \sum_{j=1}^N \left| P(V_j) - p_j \right|  \bigg) .
\end{align*}

Define $ \mathcal{G}_n = \{  (P,\Sigma) : \sum_{j=1}^{N} \left| P(V_j) - p_j     \right| \leq    \epsilon_n , \; \|  \Sigma - \Sigma_0 \| \leq  \epsilon_n \}$. The preceding bounds imply that
\begin{align*}
\int_{(P,\Sigma) : \| \Sigma - \Sigma_0  \| \leq D \epsilon_n , \|   \varphi_{ F_0^* , \Sigma_0  }  -  \varphi_{P,\Sigma_0}     \|_{L^2} \leq D \epsilon_n   }   d \nu_{\alpha,G} (P,\Sigma) \geq \int_{\mathcal{G}_n} d \nu_{\alpha,G} (P,\Sigma) .
\end{align*}

Since the prior is a product measure $\nu_{\alpha,G} = \text{DP}_{\alpha} \otimes G$, the integral appearing on the right can be expressed as
\begin{align*}
\int_{\mathcal{G}_n} d \nu_{\alpha,G} (P,\Sigma) = \int_{ \Sigma : \| \Sigma - \Sigma_0  \| \leq  \epsilon_n} \int_{P : \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n    } d  \text{DP}_{\alpha}(P)  d G(\Sigma).
\end{align*}

As $ \text{DP}_{\alpha} $ is constructed using a Gaussian base measure $\alpha $, it is straightforward to verify that $
\inf_{j=1}^N \alpha(V_j) \geq C \epsilon_n^d \exp(- C' ( \log \epsilon_n^{-1})^{2/\chi}  ) $ for universal constants $C,C' > 0$. By definition of $ \text{DP}_{\alpha}$, $(P(V_1),\dots, P(V_N)) \sim \text{Dir}(N, \alpha(V_1), \dots,  \alpha(V_N))$. As $N = D \{ \log ( \epsilon_n^{-1} )  \}^{\iota}$, an application of <cit.> implies
\begin{align*}
\int_{P : \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n    } d  \text{DP}_{\alpha}(P) \geq C \exp \big (  - C'  ( \log \epsilon_n^{-1} )^{\iota + \max \{2/ \chi,1   \}  }   \big)  & = C \exp \big (  - C'  ( \log \epsilon_n^{-1} )^{\lambda  }   \big) \\ & \geq  C \exp \big (  - C''  n \epsilon_n^2   \big)
\end{align*}

for universal constants $C,C',C'' > 0$.

It remains to bound the outer integral. The law of $G = G_n$ is given by $\Omega / \sigma_n^2$ where $\Omega \sim L$ and $L$ is a probability measure on  $\mathbf{S}_+^d$ that  satisfies Assumption <ref>. By Assumption <ref> and the definition of $\sigma_n^2$, there exists a universal constant $C,C',C'' > 0$ such that
\begin{align*}
\int_{ \Sigma : \| \Sigma - \Sigma_0  \| \leq  \epsilon_n} d G(\Sigma) = \int_{\Sigma : \| \Sigma - \sigma_n^2 \Sigma_0   \| \leq \sigma_n^2 \epsilon_n  } d L(\Sigma) \geq C  \exp \big( - C' \sigma_n^{-2 \kappa}   \big) \geq C \exp ( -C'' n \epsilon_n^2  )
\end{align*}

The  estimate for the lower bound of the normalizing constant follows from combining all the preceding bounds.

Next, we establish a preliminary local concentration bound under the prior. Observe that for any $E > 0$, we have

\begin{align*}
&  \int \limits_{ (P,\Sigma) : \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2 > 2E \epsilon_n^2 }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) \leq \exp \big( - n E \epsilon_n^2    \big).
\end{align*}

The law of  $G = G_n$ is given by $\Omega / \sigma_n^2$ where $\Omega \sim L$  and $L$ is a probability measure on  $\mathbf{S}_+^d$ that  satisfies Assumption <ref>. By Assumption <ref>, it follows that for every $E' > 0$, there exists $E > 0$ such that

\begin{align*}
\int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\Sigma) = \int_{\Sigma : \| \Sigma^{-1}  \| > E (n \epsilon_n^2)^{1/ \kappa}  }   d L(\Sigma) \leq \exp \big(  - E' n \epsilon_n^2  \big).
\end{align*}

As the prior is a product measure $\nu_{\alpha,G} = \text{DP}_{\alpha} \otimes G$ and  $ \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2  \geq 0$, the preceding bound implies
\begin{align*}
&  \int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma)  \\ & \leq  \int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\Sigma)  \\ & \leq \exp \big(  - E' n \epsilon_n^2  \big)..
\end{align*}

From combining the preceding bounds, it follows that for any $E' > 0$ we can pick $E > 0$ sufficiently large such that
\begin{align*}
\int \limits_{ (P,\Sigma) : \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2 \leq E \epsilon_n^2, \| \Sigma^{-1} \| \leq E \sigma_n^2 (n \epsilon_n^2)^{1/\kappa} }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T_n)}^2 &        \bigg)    d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \exp \big( - E' n \epsilon_n^2   \big) .
\end{align*}

We prove the main statement of the theorem. From the bounds derived in steps $(i)$ and $(ii)$, it follows that for any $C' > 0$, there exists a $ M > 0$ such that

\begin{align*}
\nu_{\alpha,G} \bigg(  \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2 \leq M^2 \epsilon_n^2, \| \Sigma^{-1} \| \leq M^2 \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg ) \geq 1- \exp(-C' n \epsilon_n^2)
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

For any choice of $(P,\Sigma)$ satisfying  $\| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)} \leq M \epsilon_n$, an application of Lemma <ref> (and noting that $ \left| \varphi_{P,\Sigma} \right|  \leq 1$) yields
\begin{align*}
\| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}   & \leq \| {\varphi}_{Y} - \widehat{\varphi_{Y}}    \|_{\mathbb{B}(T_n)} + \|  ({\varphi}_{\epsilon} - \widehat{\varphi_{\epsilon}} ) \varphi_{P,\Sigma}   \|_{\mathbb{B}(T_n)} + \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}  \\ & \leq  \| {\varphi}_{Y} - \widehat{\varphi_{Y}}    \|_{\mathbb{B}(T_n)} + \|  {\varphi}_{\epsilon} - \widehat{\varphi_{\epsilon}}   \|_{\mathbb{B}(T_n)} + \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)} \\ & \leq D \frac{\sqrt{ T_n^d \log T_n}}{\sqrt{n}}   +  \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)} \\ & \leq D \epsilon_n +  \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)} \\ & \leq D \epsilon_n.
\end{align*}

Let $\uptau_{T}  = \sup_{\| t \|_{\infty} \leq T } \left|  \varphi_{\epsilon} (t)  \right|^{-1} $. Since $ \varphi_Y = \varphi_X \varphi_{\epsilon} $, the preceding bound implies that

\begin{align*}
D \epsilon_n \geq  \| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)} & = \| \big( {\varphi}_{X}   -   {\varphi}_{P,\Sigma} \big) \varphi_{\epsilon}      \|_{\mathbb{B}(T_n)} \\ & \geq   \uptau_{T_n}^{-1} \| \big( {\varphi}_{X}   -   {\varphi}_{P,\Sigma} \big) \mathbbm{1} \{  t \in \mathbb{B}(T_n)  \}   \|_{L^2}.
\end{align*}

It remains to examine the bias from truncating the $L^2$ norm to the set $\mathbb{B}(T_n)$. Suppose $ \|  \Sigma^{-1}  \| \leq M^2  \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}  $ holds. It follows that there exists a $c > 0$ for which $  \lambda_1(\Sigma ) \geq c (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} $ holds. From the definition of $\sigma_n^2$, we have  $T_n^2 (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} \asymp  (\log n) ( \log \log n )  $ in the mildly ill-posed case. In the severely ill-posed case, we have
\begin{align*}
T_n^2 (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} \asymp \begin{cases}  (\log n) ( \log \log n) & \zeta \in (0,2) \\ \log n & \zeta=2.  \end{cases}
\end{align*}

It follows that there exists a universal constant $C > 0$ such that

\begin{align*}
\| \big( {\varphi}_{X}   -   {\varphi}_{P,\Sigma} \big) \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 & \leq  2 \|  {\varphi}_{X}    \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 +  2 \|     {\varphi}_{P,\Sigma}  \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 \\ & \leq 2 \int_{\| t \|_{\infty} > T_n}  e^{- t' \Sigma_0 t} dt + 2 \int_{\| t \|_{\infty} > T_n} e^{- t' \Sigma t} dt \\ & \leq 2 \int_{\| t \|_{\infty} > T_n}  e^{- t' \Sigma_0 t} dt + 2 \int_{\| t \|_{\infty} > T_n} e^{- c \| t \|^2  \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa}   } dt \\ & \leq D \bigg[  e^{- C T_n^2} T_n^{d-2} +  \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa}   e^{- C T_n^2 \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa} } T_n^{d-2}   \bigg ] .
\end{align*}

For all mildly ill-posed models and severely ill-posed with $\zeta \in (0,2)$, the preceding bound  reduces to $Dn^{-1}$. For severely ill-posed models with $\zeta =2$  it reduces to $D n^{- 2 K  }$ for some constant $ K \in (0,1/2]$ (that depends on, among other factors, the smallest eigenvalue of $\Sigma_0$).

We verify the conclusion of the theorem. Suppose that the model is mildly ill-posed or severely ill-posed with $\zeta \in (0,2)$. From combining the preceding bounds (and noting that  $ n^{-1} \lessapprox \uptau_{T_n} \epsilon_n $), it follows that for every $C' > 0$, there exists a $ M > 0$ such that
$$ \nu_{\alpha,G} \bigg(  \| \varphi_{X}  -  \varphi_{P,\Sigma}      \|_{L^2} \leq M \uptau_{T_n} \epsilon_n \:     \big|\mathcal{Z}_n,T_n  \bigg) \geq 1- \exp(-C' n \epsilon_n^2) . $$

holds with $\mathbb{P}$ probability approaching $1$. If the model is severely ill-posed with constant $K$ as specified above, we have
$$ \nu_{\alpha,G} \bigg(  \| \varphi_{X}  -  \varphi_{P,\Sigma}      \|_{L^2} \leq M \uptau_{T_n} \epsilon_n + n^{-K}  \:     \big|\mathcal{Z}_n,T_n  \bigg) \geq 1- \exp(-C' n \epsilon_n^2)  $$

holds with $\mathbb{P}$ probability approaching $1$.

For mildly ill-posed models, the claim follows from observing that
\begin{align*}
\uptau_{T_n} \epsilon_n \asymp \frac{(\log n)^{(\lambda + \zeta)/2} }{\sqrt{n}} (\log \log n)^{\zeta/2}.
\end{align*}

Similarly, for severely ill-posed models with $\zeta \in (0,2]$ and $T_n =  (c_0 \log n)^{1/ \zeta} $ for some $c_0$ satisfying $c_0 R  = \gamma < 1/2$, we have
$$ \uptau_{T_n} \epsilon_n \asymp n^{-1/2} (\log n)^{\lambda/2}  e^{\gamma \log n }   =n^{\gamma -1/2} (\log n)^{\lambda/2} .    $$

The analysis is analogous to that of Theorem <ref>. Let $\uptau_{T}  = \sup_{\| t \|_{\infty} \leq T } \left|  \varphi_{\epsilon} (t)  \right|^{-1} $. From the proof of that result, it follows that for any $C' > 0$, there exists a $M > 0$ such that
\begin{align*}
\nu_{\alpha,G} \bigg(  \| \varphi_{X}    -  \varphi_{P,\Sigma}     \|_{\mathbb{B}(T_n)}^2 \leq M^2 \uptau_{T_n}^2 \epsilon_n^2, \| \Sigma^{-1} \| \leq M^2 \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg ) \geq 1- \exp(-C' n \epsilon_n^2)
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$. For any $(P,\Sigma)$ an application of Cauchy-Schwarz yields
\begin{align*}
\int_{\mathbb{B}(T_n)} \left|  \varphi_{X} (t) - \varphi_{P,\Sigma} (t)    \right| dt \leq  D T_n^{d/2}   \| \varphi_{X}    -  \varphi_{P,\Sigma}     \|_{\mathbb{B}(T_n)}.
\end{align*}

In particular, convergence rates for the quantity on the right imply convergence rates for the left up to a multiplicative factor of $ T_n^{d/2}$. It remains to examine the bias from truncating the $L^1$ norm to the set $\mathbb{B}(T_n)$. Suppose $ \|  \Sigma^{-1}  \| \leq M^2  \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}  $ holds. It follows that there exists a $c > 0$ for which $  \lambda_1(\Sigma ) \geq c (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} $ holds. From the definition of $\sigma_n^2$, we have  $T_n^2 (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} \asymp  (\log n) ( \log \log n )  $ in the mildly ill-posed case. In the severely ill-posed case, we have
\begin{align*}
T_n^2 (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} \asymp \begin{cases}  (\log n) ( \log \log n) & \zeta \in (0,2) \\ \log n & \zeta=2.  \end{cases}
\end{align*}

It follows that there exists a universal constant $C > 0$ such that

\begin{align*}
\| \big( {\varphi}_{X}   -   {\varphi}_{P,\Sigma} \big) \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^1} & \leq   \|  {\varphi}_{X}    \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^1} +   \|     {\varphi}_{P,\Sigma}  \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^1} \\ & \leq  \int_{\| t \|_{\infty} > T_n}  e^{- t' \Sigma_0 t/2} dt +  \int_{\| t \|_{\infty} > T_n} e^{- t' \Sigma t/2} dt \\ & \leq  \int_{\| t \|_{\infty} > T_n}  e^{- t' \Sigma_0 t/2} dt +  \int_{\| t \|_{\infty} > T_n} e^{- c \| t \|^2  \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa} / 2   } dt \\ & \leq D \bigg[  e^{- C T_n^2} T_n^{d-2} +  \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa}   e^{- C T_n^2 \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa} } T_n^{d-2}   \bigg ] .
\end{align*}

For all mildly ill-posed models and severely ill-posed with $\zeta \in (0,2)$, the preceding bound reduces to $Dn^{-1/2}$. For severely ill-posed models with $\zeta =2$  it reduces to $D n^{-  K  }$ for some constant $ K \in (0,1/2]$ (that depends on, among other factors, the smallest eigenvalue of $\Sigma_0$).

From combining the preceding bounds, we obtain (similarly to the concluding remarks in the proof of Theorem <ref>) contraction rates for $\| \varphi_{X} - \varphi_{P,\Sigma} \|_{L^1}$. The claim then follows from observing that $ \| f_X - \phi_{P,\Sigma}  \|_{L^{\infty}} \leq  \| \varphi_{X} - \varphi_{P,\Sigma} \|_{L^1}. $

The proof proceeds through several steps which we outline below. We use $D > 0$ as a generic universal constant that may change from line to line. For ease of notation, we suppress the dependence of $m = m_n$ and $G = G_n$ on $n$.

\begin{align*}
\alpha_n = \epsilon_n^{1/(p + \zeta)} \; \; , \; \;  \epsilon_n^2 = \frac{(\log n)^{\lambda + d/2 } }{n^{ \frac{2(p + \zeta)}{2(p + \zeta) + d}}  }          \; \; \; \;   ,  \; \; \; \;  \lambda = \begin{cases}
\chi^{-1}(d +2)  & \chi < 2 \\ d/ \chi + 1 & \chi \geq 2.
\end{cases}
\end{align*}

First, we derive a lower bound for the normalizing constant of the posterior measure.  Specifically, we aim to show that there exists a $C > 0 $ such that
\begin{align}
\label{t1lb} \int  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T_n)  }^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) \geq \exp \big(   - C n    \epsilon_{n}^2   \big)
\end{align}

holds with $\mathbb{P}$ probability approaching $1$.

Since $m = m_n \asymp n$, an application of Lemma <ref> implies that
\begin{align*}
&  \| \widehat{\varphi}_{Y}  - \varphi_{Y}  \|_{\mathbb{B}(T_n)}^2   =  \int_{\mathbb{B}(T_n)}   \left|  \widehat{\varphi}_Y(t) - \varphi_Y(t)    \right|^2 dt \leq D \frac{ T_n^d  \log(T_n)  }{n}  \; ,  \\ & \| \widehat{\varphi}_{\epsilon}  - \varphi_{\epsilon}  \|_{\mathbb{B}(T_n)}^2 =  \int_{\mathbb{B}(T_n)}   \left|  \widehat{\varphi}_\epsilon(t) - \varphi_\epsilon(t)    \right|^2 dt \leq D \frac{ T_n^d \log(T_n)  }{n} .
\end{align*}

with $\mathbb{P}$ probability approaching $1$. On this set (and using that $ \left| \varphi_{P,\Sigma} \right|  \leq 1$ ) we obtain
\begin{align*}
\| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2  & \leq D \bigg( \| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2    +  \frac{ T_n^d \log (T_n)}{n}     \bigg) \\ & \leq D \bigg(   \| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2  + \epsilon_n^2  \bigg) \; ,
\end{align*}

where the last inequality follows from the definition of $T_n$ and $\epsilon_n$. It follows that

\begin{align*}
& \int  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma)  \\ & \geq \exp(-n D \epsilon_n^2 )   \int  \exp \bigg( - nD\| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) .
\end{align*}

By Condition <ref>, there exists universal constants $ \chi , C, M    < \infty$  and  a mixing distribution $S_{\alpha_n}$ supported on the cube $ I_{n} = [-C  (\log \epsilon_n^{-1})^{1/\chi},C  (\log \epsilon_n^{-1})^{1/\chi} ]^d$ that satisfies
$$ \| f_{X} \star f_{\epsilon}   -  \phi_{S_{\alpha_n},\alpha_n^2 I} \star f_{\epsilon}  \|_{L^2} \leq D  \epsilon_n^2 .   $$

Note that $ \F[ f_X \star f_{\epsilon} ]  = \varphi_{X} \varphi_{\epsilon}  $ and $\F[  \phi_{S_{\alpha_n},\alpha_n^2 I} \star f_{\epsilon} ] =  \varphi_{\epsilon} \varphi_{S_{\alpha_n}, \alpha_n^2 I}  $. As the Fourier transform preserves $L^2$ distance (up to a constant), it follows that

\begin{align*}
\|  \varphi_Y - \varphi_{\epsilon} \varphi_{S_{\alpha_n}, \alpha_n^2 I }       \|_{\mathbb{B}(T_n)}^2 \leq  D  \|  \varphi_Y - \varphi_{\epsilon} \varphi_{S_{\alpha_n}, \alpha_n^2 I}       \|_{L^2}^2 & = D   \|  \varphi_{X} \varphi_{\epsilon}  - \varphi_{\epsilon} \varphi_{S_{\alpha_n}, \alpha_n^2 I}       \|_{L^2}^2  \\ & \leq D  \| f_{X} \star f_{\epsilon}   -  \phi_{S_{\alpha_n},\alpha_n^2 I} \star f_{\epsilon}  \|_{L^2}^2  \\ & \leq D \alpha_n^{2(p + \zeta)}  \\ & \leq D \epsilon_n^2.
\end{align*}

From combining the preceding bounds, we obtain

\begin{align*}
& \int  \exp \bigg( - nD \| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2     \bigg) d \nu_{\alpha,G} (P,\Sigma)  \\ & \geq \exp \big( - n D  \epsilon_{n}^2   \big) \int \exp \bigg( - nD \| \varphi_{\epsilon} \varphi_{S_{\alpha_n}, \alpha_n^2 I}     - {\varphi}_{\epsilon} {\varphi}_{P, \Sigma}      \|_{\mathbb{B}(T_n)}^2          \bigg) d \nu_{\alpha,G} (P,\Sigma).
\end{align*}

Fix any $(P,\Sigma)$ in the support of $\nu_{\alpha,G} $. Since $\left| \varphi_{\epsilon}   \right| \leq 1$, we have that

\begin{align*}
\| \varphi_{\epsilon} \varphi_{S_{\alpha_n}, \alpha_n^2 I }     - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)} \leq \|  \varphi_{S_{\alpha_n}, \alpha_n^2 I}     - {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}  .
\end{align*}

By an application of Lemma <ref>, there exists a discrete probability measure $ F_{\alpha_n} = \sum_{i=1}^N p_i \delta_{\mu_i}  $ with $N \leq D T_n^d \{\log( \epsilon_n^{-1}) \}^{ d/ \chi }$  and  $\mu_i \in I_{n}$  that satisfies

$$ \|   \varphi_{S_{\alpha_n}, \alpha_n^2 I}   - \varphi_{F_{\alpha_n}, \alpha_n^2 I}       \|_{\mathbb{B}(T_n)} \leq D \epsilon_n .$$

From the second claim of Lemma <ref>, we can also assume without loss of generality that the support points have separation satisfying $\inf_{k \neq j} \| \mu_k  - \mu_j \| \geq   \epsilon_n T_n^{-(d+2)/2} $. From combining the preceding bounds, it follows that
\begin{align*}
&  \int \exp \bigg( - nD \| \varphi_{\epsilon} \varphi_{S_{\alpha_n}, \alpha_n^2 I}     - {\varphi}_{\epsilon} {\varphi}_{P, \Sigma}      \|_{\mathbb{B}(T_n)}^2          \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \exp(-n D \epsilon_n^2 )   \int \exp \bigg( - nD \|  \varphi_{F_{\alpha_n}, \alpha_n^2 I}     -  {\varphi}_{P, \Sigma}      \|_{\mathbb{B}(T_n)}^2          \bigg) d \nu_{\alpha,G} (P,\Sigma).
\end{align*}

Define the set
\begin{align*}
\Omega_n =  \bigg \{ \Sigma \in \mathbf{S}_{+}^d  : \lambda_j(\Sigma) \in \bigg[  \frac{ \alpha_n^2 }{1 + \epsilon_n T_n^{-(d+2)/2}  }  ,      \alpha_n^2   \bigg ]  \; \; \forall \; j=1,\dots,d.   \bigg \}
\end{align*}

Observe that for any distribution $P$ and  $\Sigma \in \Omega_n$, an application of Lemma <ref> yields
\begin{align*}
\|  \varphi_{P, \alpha_n^2 I}     -  {\varphi}_{P, \Sigma}   \|_{\mathbb{B}(T_n)} \leq D T_n^{(d+2)/2} \|  \Sigma -  \alpha_n^2 I  \|  & = D T_n^{(d+2)/2} \max_{j=1,\dots,d} \left|   \lambda_j(\Sigma) - \alpha_n^2   \right| \\ & \leq D \epsilon_n.
\end{align*}

It follows that
\begin{align*}
&  \int \exp \bigg( - nD \|  \varphi_{F_{\alpha_n}, \alpha_n^2 I}     -  {\varphi}_{P, \Sigma}      \|_{\mathbb{B}(T_n)}^2          \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \int_{\Sigma \in \Omega_n} \exp \bigg( - nD \|  \varphi_{F_{\alpha_n}, \alpha_n^2 I}     -  {\varphi}_{P, \Sigma}      \|_{\mathbb{B}(T_n)}^2          \bigg) d \nu_{\alpha,G} (P,\Sigma)  \\ & \geq \exp(-nD \epsilon_n^2) \int_{\Sigma \in \Omega_n} \exp \bigg( - nD \|  \varphi_{F_{\alpha_n}, \alpha_n^2 I}     -  {\varphi}_{P, \alpha_n^2 I }      \|_{\mathbb{B}(T_n)}^2          \bigg) d \nu_{\alpha,G} (P,\Sigma).
\end{align*}

Define $V_i = \{ t \in \R^d : \| t - \mu_i \| \leq  \epsilon_n^2 T_n^{-(d+2)/2} \} $ for $i=1,\dots,N$ and $V_0 = \R^d \setminus \bigcup_{i=1}^N V_i $. From the definition of the $\{\mu_i \}_{i=1}^N$, it follows that $\{V_0, V_1, \dots , V_N  \}$ is a disjoint partition of $\R^d$. For any fixed distribution $P$ an application of  Lemma <ref> yields
\begin{align*}
\|  \varphi_{P, \alpha_n^2 I} - \varphi_{ F_{\alpha_n} , \alpha_n^2 I }  \|_{\mathbb{B}(T_n)} & \leq D \bigg[ \epsilon_n  + T_n^{d/2} \sum_{j=1}^N \left| P(V_j) - p_j \right|  \bigg] .
\end{align*}

The preceding bound implies that
\begin{align*}
&  \int_{\Sigma \in \Omega_n} \exp \bigg( - nD \|  \varphi_{F_{\alpha_n}, \alpha_n^2 I}     -  {\varphi}_{P, \alpha_n^2 I }      \|_{\mathbb{B}(T_n)}^2          \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ & \geq  \int_{ (P,\Sigma) : \Sigma  \in \Omega_n , \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n T_n^{-d/2}   } \exp \bigg( - nD \|  \varphi_{F_{\alpha_n}, \alpha_n^2 I}     -  {\varphi}_{P, \alpha_n^2 I }      \|_{\mathbb{B}(T_n)}^2          \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \exp(-n D \epsilon_n^2 ) \int_{ (P,\Sigma) : \Sigma  \in \Omega_n , \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n  T_n^{-d/2}   }  d \nu_{\alpha,G} (P,\Sigma) .
\end{align*}

Since the prior is a product measure $\nu_{\alpha,G} = \text{DP}_{\alpha} \otimes G$, the integral appearing on the right can be expressed as
\begin{align*}
\int_{ \Sigma \in \Omega_n }\int_{P : \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n T_n^{-d/2}    } d  \text{DP}_{\alpha}(P)  d G(\Sigma).
\end{align*}

As $ \text{DP}_{\alpha} $ is constructed using a Gaussian base measure $\alpha $, it is straightforward to verify that $
\inf_{j=1}^N \alpha(V_j) \geq C  \epsilon_n^{2d} T_n^{-d(d+2)/2}   \exp(- C' ( \log \epsilon_n^{-1})^{2/\chi}  ) $ for universal constants $C,C' > 0$.  By definition of $ \text{DP}_{\alpha}$, $(P(V_1),\dots, P(V_N)) \sim \text{Dir}(N, \alpha(V_1), \dots,  \alpha(V_N))$. As $N = D T_n^d \{\log( \epsilon_n^{-1}) \}^{ d/ \chi }$, an application of <cit.> and the definition of $(T_n,\epsilon_n^2)$ implies
\begin{align*}
\int_{P : \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n T_n^{-d/2}    } d  \text{DP}_{\alpha}(P) & \geq   C \exp \big (  - C'  T_n^d  \{   \log \epsilon_n^{-1} \}^{ d/\chi +   \max \{2/ \chi,1   \}  }   \big) \\ &          \geq C \exp(-C' n \epsilon_n^2).
\end{align*}

It remains to bound the outer integral. The law of $G= G_n$ is given by $\Omega / \sigma_n^2$ where $\Omega \sim L$ and $L$ is a probability measure on  $\mathbf{S}_+^d$ that  satisfies Assumption <ref>. By Assumption <ref> and the definition of $(\alpha_n^2,\sigma_n^2, \epsilon_n^2)$, there exists a universal constant $C,C',C'' > 0$ such that
\begin{align*}
\int_{  \Sigma \in \Omega_n } d G(\Sigma) = \int_{  \Sigma \in \sigma_n^2 \Omega_n } d L(\Sigma) \geq C \exp \big( - C' \sigma_n^{-2 \kappa} \alpha_n^{-2 \kappa}   \big) \geq C \exp ( -C'' n \epsilon_n^2  ).
\end{align*}

The  estimate for the lower bound of the normalizing constant follows from combining all the preceding bounds.

Next, we establish a preliminary local concentration bound under the prior. Observe that for any $E > 0$, we have

\begin{align*}
&  \int \limits_{ (P,\Sigma) : \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2 > 2E \epsilon_n^2 }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) \leq \exp \big( - n E \epsilon_n^2    \big).
\end{align*}

The law of $G= G_n$ is given by $\Omega / \sigma_n^2$ where $\Omega \sim L$ and $L$ is a probability measure on  $\mathbf{S}_+^d$ that  satisfies Assumption <ref>. By Assumption <ref>, it follows that for every $E' > 0$, there exists $E > 0$ such that

\begin{align*}
\int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\Sigma) = \int_{\Sigma : \| \Sigma^{-1}  \| > E (n \epsilon_n^2)^{1/ \kappa}  }   d L(\Sigma) \leq \exp \big(  - E' n \epsilon_n^2  \big).
\end{align*}

As the prior is a product measure $\nu_{\alpha,G} = \text{DP}_{\alpha} \otimes G$ and  $ \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2  \geq 0$, the preceding bound implies
\begin{align*}
&  \int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T_n)}^2        \bigg) d \nu_{\alpha,G} (P,\Sigma)  \\ & \leq  \int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\Sigma)  \\ & \leq \exp \big(  - E' n \epsilon_n^2  \big)..
\end{align*}

From combining the preceding bounds, it follows that for any $E' > 0$ we can pick $E > 0$ sufficiently large such that
\begin{align*}
\int \limits_{ (P,\Sigma) : \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2 \leq E \epsilon_n^2, \| \Sigma^{-1} \| \leq E \sigma_n^2 (n \epsilon_n^2)^{1/\kappa} }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{ \mathbb{B}(T_n)}^2 &        \bigg)    d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \exp \big( - E' n \epsilon_n^2   \big) .
\end{align*}

We prove the main statement of the theorem. From the bounds derived in steps $(i)$ and $(ii)$, it follows that for any $C' > 0$, there exists a $ M > 0$ such that

\begin{align*}
\nu_{\alpha,G} \bigg(  \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2 \leq M^2 \epsilon_n^2, \| \Sigma^{-1} \| \leq M^2 \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg ) \geq 1- \exp(-C' n \epsilon_n^2)
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

For any choice of $(P,\Sigma)$ satisfying  $\| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)} \leq M \epsilon_n$, an application of Lemma <ref> (and noting that $ \left| \varphi_{P,\Sigma} \right|  \leq 1$) yields
\begin{align*}
\| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}   & \leq \| {\varphi}_{Y} - \widehat{\varphi_{Y}}    \|_{\mathbb{B}(T_n)} + \|  ({\varphi}_{\epsilon} - \widehat{\varphi_{\epsilon}} ) \varphi_{P,\Sigma}   \|_{\mathbb{B}(T_n)} + \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}  \\ & \leq  \| {\varphi}_{Y} - \widehat{\varphi_{Y}}    \|_{\mathbb{B}(T_n)} + \|  {\varphi}_{\epsilon} - \widehat{\varphi_{\epsilon}}   \|_{\mathbb{B}(T_n)} + \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)} \\ & \leq D \frac{\sqrt{ T_n^d \log T_n}}{\sqrt{n}}   +  \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)} \\ & \leq D \epsilon_n +  \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)} \\ & \leq D \epsilon_n.
\end{align*}

Since $ \varphi_Y = \varphi_X \varphi_{\epsilon} $ and $\inf_{ \|t \|_{\infty} \leq T_n} \left| \varphi_{\epsilon}(t)  \right| \geq \uptau_{T_n}^{-1} $, the preceding bound implies that

\begin{align*}
D \epsilon_n \geq  \| {\varphi}_{Y}    - {\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)} & = \| \big( {\varphi}_{X}   -   {\varphi}_{P,\Sigma} \big) \varphi_{\epsilon}      \|_{\mathbb{B}(T_n)} \\ & \geq   \uptau_{T_n}^{-1} \| \big( {\varphi}_{X}   -   {\varphi}_{P,\Sigma} \big) \mathbbm{1} \{  t \in \mathbb{B}(T_n)  \}   \|_{L^2}.
\end{align*}

It remains to examine the bias from truncating the $L^2$ norm to the set $\mathbb{B}(T_n)$. Suppose $ \|  \Sigma^{-1}  \| \leq M^2  \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}  $ holds. It follows that there exists a $c > 0$ for which $  \lambda_1(\Sigma ) \geq c (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} $ holds. From the definition of $\sigma_n^2$, we have  $T_n^2 (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} \asymp  (\log n) ( \log \log n )  $.

Since $ f_X \in \mathbf{H}^p (M) $, we have that
\begin{align*}
\int_{\|t \|_{\infty} > T_n} \left|  \varphi_X(t)  \right|^2 dt \leq D T_n^{-2p}.
\end{align*}

It follows that there exists a universal constant $C > 0$ such that

\begin{align*}
\| \big( {\varphi}_{X}   -   {\varphi}_{P,\Sigma} \big) \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 & \leq  2 \|  {\varphi}_{X}    \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 +  2 \|     {\varphi}_{P,\Sigma}  \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 \\ & \leq 2 \int_{\|t \|_{\infty} > T_n} \left|  \varphi_X(t)  \right|^2 dt + 2 \int_{\| t \|_{\infty} > T_n} e^{- t' \Sigma t} dt \\ & \leq 2 \int_{\|t \|_{\infty} > T_n} \left|  \varphi_X(t)  \right|^2 dt + 2 \int_{\| t \|_{\infty} > T_n} e^{- c \| t \|^2  \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa}   } dt \\ & \leq D \bigg[  T_n^{-2p} +  \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa}   e^{- C T_n^2 \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa} } T_n^{d-2}   \bigg ] .
\end{align*}

Since $T_n^2 (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} \asymp  (\log n) ( \log \log n )  $, the preceding bound reduces to $DT_n^{-2p}$. From combining the preceding bounds (and noting that  $ T_n^{-p} \lessapprox \uptau_{T_n} \epsilon_n $), it follows that for every $C' > 0$, there exists a $ M > 0$ such that
$$ \nu_{\alpha,G} \bigg(  \| \varphi_{X}  -  \varphi_{P,\Sigma}      \|_{L^2} \leq M \uptau_{T_n} \epsilon_n \:     \big|\mathcal{Z}_n,T_n  \bigg) \geq 1- \exp(-C' n \epsilon_n^2) . $$

holds with $\mathbb{P}$ probability approaching $1$. The claim follows from observing that
\begin{align*}
\uptau_{T_n} \epsilon_n \asymp T_n^{\zeta} \epsilon_n \asymp n^{-p/[2(p + \zeta) + d]} (\log n)^{(\lambda + \zeta)/2 + d/4}.
\end{align*}

The proof proceeds through several steps which we outline below. We use $D > 0$ as a generic universal constant that may change from line to line.
\begin{align*}
\epsilon_n^2 = n^{-1} (\log n)^{\lambda} \; \; \; \; ,  \; \; \; \;  \lambda = \begin{cases}
\chi^{-1}(d +2) + d/2 & \chi < 2 \\ d + 1 & \chi \geq 2.
\end{cases}
\end{align*}

First, we derive a lower bound for the normalizing constant of the posterior measure.  Specifically, we aim to show that there exists a $C > 0 $ such that
\begin{align}
\label{t1lb}  \int  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \geq \exp \big(   - C n    \epsilon_{n}^2   \big)
\end{align}

holds with $\mathbb{P}$ probability approaching $1$.

Fix  $\epsilon > 0$ sufficiently small. By Condition <ref>, the mixing distribution $F_0$ satisfies $F_0( t \in \R^d :  \| t \|  > z) \leq C \exp(- C' z^{\chi})$ for some $C,C' > 0$. Fix a universal constant $R >0$ such that $\exp(- C' z^{\chi}) \leq \epsilon $ for every $z > R (\log \epsilon^{-1})^{1/\chi}$. In particular, note that the cube $I =[-R  (\log \epsilon^{-1})^{1/\chi},R  (\log \epsilon^{-1})^{1/\chi} ]^d $ satisfies $ 1- F_0(I) \leq D \epsilon$. Denote the probability measure induced from the restriction of $F_0$ to $I$ by
$$ \overline{F}_0(A) = \frac{F_0(A \cap I)}{F_0(I)}  \; \; \; \; \; \forall \; \; \text{Borel}  \; A \subseteq \R^d. $$

Observe that
\begin{align*}
\sup_{t \in \R^d} \left| \varphi_{F_0} (t) -  \varphi_{\overline{F}_0}(t) \right| =  \sup_{t \in \R^d} \left|  \int_{\R^d} e^{\mathbf{i} t' x} d(F - \overline{F}_0 ) (x)     \right|   \leq \| F_0 - \overline{F}_0   \|_{TV}  \leq 1- F_0(I) \leq D \epsilon.
\end{align*}

For all sufficiently large $ M > 0$, the tail bound on the mixing distribution $F_0$ implies

\begin{align*}
\E[ \| X \| \mathbbm{1} \{ \| X \| > M   \}  ]  &  = \int_{0}^{\infty} \mathbb{P} \big(  \| X \| > M, \| X \| > t      \big) dt \\ & \leq   M \mathbb{P} \big( \| X \| > M     \big)  + \int_{M}^{\infty} \mathbb{P} \big(  \| X \| > t \big) dt       \\ & \leq D \bigg[ M  \exp(- C' M^{\chi})  + \int_M^{\infty} \exp(- C' t^{\chi})      dt               \bigg]  \\ & \leq D \bigg[ M  \exp(- C' M^{\chi}) + \exp(- C' M^{\chi}) M^{1- \chi}   \bigg].
\end{align*}

It follows that

\begin{align*}  \sup_{t \in \R^d} \| \nabla  \varphi_{F_0}(t) -  \nabla \varphi_{\overline{F}_0}(t) \| &  =  \sup_{t \in \R^d} \bigg \|  \int_{\R^d} x e^{\mathbf{i} t' x} d(F_0 - \overline{F}_0 ) (x)     \bigg \|  \\ & \leq \sup_{t \in \R^d}  \bigg\|  \int_{ x \in I} x e^{\mathbf{i} t' x} d(F_0- \overline{F}_0)  \bigg \| + \sup_{t \in \R^d} \bigg \| \int_{x \notin I} x e^{\mathbf{i} t' x} d F_0   \bigg \|  \\ & \leq D \bigg[ (\log \epsilon^{-1})^{1/\chi} \| F_0 - \overline{F}_0   \|_{TV} +  (\log \epsilon^{-1})^{1/\chi} \epsilon      \bigg] \\  & \leq D (\log \epsilon^{-1})^{1/\chi} \epsilon.
\end{align*}

By Condition <ref>, $ \inf_{\| t \|_{\infty} \leq T_n }  \left| \varphi_{F_0}(t) \right| \geq c \exp(-c' T_n^2 ) $ for some $c,c' > 0$. Since $T_n^2 \lessapprox \log(n) $, it follows that there exists a $C_1 > 0$ such that
$$ \inf_{\| t \|_{\infty} \leq T_n }  \left| \varphi_{F_0}(t) \right| \geq n^{-C_1} .$$

In particular, we can choose $L > 1$ large enough such that the choice $ \epsilon  = \epsilon_n^L $ implies $ \overline{F}_0 $  has support contained in the cube $I_n =[-E  (\log \epsilon_n^{-1})^{1/\chi},E (\log \epsilon_n^{-1})^{1/\chi} ]^d $  for some universal constant $E > 0$
and $  T_n n^{C_1} (\log \epsilon_n^{-L})^{1/\chi}  \epsilon_n^L \leq \epsilon_n  $.  Since $ \sup_{\| t \|_{\infty} \leq T_n}  \| \nabla \log \varphi_{F_0}  (t) \|  \lessapprox T_n $, it follows that

\begin{align*}
&  \sup_{ \| t \|_{\infty} \leq T_n} \| \nabla \log \varphi_{F_0}  (t) \| \frac{\left|  \varphi_{F_0} (t) -  \varphi_{\overline{F}_0}(t)  \right|}{\left| \varphi_{F_0} (t) \right|} \leq  \sup_{\| t \|_{\infty} \leq T_n }  \frac{\| \nabla \log \varphi_{F_0}  (t)  \|}{\left| \varphi_{F_0}(t) \right| }     \sup_{t \in \R^d} \left| \varphi_{F_0}(t) - \varphi_{\overline{F}_0}(t) \right|     \leq   D \epsilon_n   \\ &  \sup_{ \| t \|_{\infty}  \leq T_n} \frac{ \| \nabla \varphi_{F_0}(t) -  \nabla \varphi_{\overline{F}_0}(t)\| }{ \left|  \varphi_{F_0}(t)  \right|} \leq \sup_{\| t \|_{\infty} \leq T_n }  \left| \varphi_{F_0}(t) \right|^{-1} \sup_{t \in \R^d}  \| \nabla  \varphi_{F_0}(t) -  \nabla \varphi_{\overline{F}_0}(t) \| \leq D \epsilon_n.
\end{align*}

Since $\epsilon_n \downarrow 0$, observe that the preceding bound also implies
$$  \left|   \varphi_{\overline{F}_0}(t)  \right| \geq \left| \varphi_{F_0}(t)    \right| - \left| \varphi_{F_0}(t) - \varphi_{\overline{F}_0}(t) \right| \geq   \left| \varphi_{F_0}(t)    \right| - D \epsilon_n \left| \varphi_{F_0}(t)    \right|  \geq \frac{1}{2} \left| \varphi_{F_0}(t)    \right|   $$

for all sufficiently large $n$.

It follows that
\begin{align*}
& \sup_{\| t \|_{\infty} \leq T_n}  \| \nabla \log  \varphi_{F_0}  (t) - \nabla \log \varphi_{\overline{F}_0}(t)     \| \\ & = \sup_{\| t \|_{\infty} \leq T_n}  \bigg \|  \nabla \log \varphi_{F_0} (t) \frac{\varphi_{\overline{F}_0} (t) - \varphi_{F_0} (t) }{ \varphi_{\overline{F}_0}(t) }     +  \frac{ \nabla  \varphi_{F_0} (t)  - \nabla  \varphi_{\overline{F}_0} (t)  }{\varphi_{\overline{F}_0}(t)  }         \bigg \| \\ & \leq \sup_{\| t \|_{\infty} \leq T_n} \bigg(  \left|  \frac{\varphi_{\overline{F}_0}(t) - \varphi_{F_0}(t) }{ \varphi_{\overline{F}_0} (t) }    \right| \|  \nabla \log  \varphi_{F_0} (t)  \| \: +  \: \bigg \|  \frac{ \nabla  \varphi_{F_0} (t)  - \nabla  \varphi_{\overline{F}_0} (t)  }{\varphi_{\overline{F}_0}(t)  }     \bigg \|        \bigg)  \\ & \leq  2 \sup_{\| t \|_{\infty} \leq T_n} \bigg(  \left|  \frac{\varphi_{\overline{F}_0}(t) - \varphi_{F_0}(t) }{ \varphi_{{F}_0} (t) }    \right| \|  \nabla \log  \varphi_{F_0} (t)  \| \: +  \: \bigg \|  \frac{ \nabla  \varphi_{F_0} (t)  - \nabla  \varphi_{\overline{F}_0} (t)  }{\varphi_{{F}_0}(t)  }     \bigg \|        \bigg) \\ & \leq D \epsilon_n.
\end{align*}

Next, we show that $ \overline{F}_0  $ can be suitably approximated by a discrete measure. Let $\iota = \max \{ d, d/\chi + d/2   \} $.  By Lemma <ref>, there exists a discrete measure $F' = \sum_{i=1}^N p_i \delta_{\mu_i} $ with at most $ N = D  (\log \epsilon_n^{-1})^{\iota} $ support points on $I_n$ such that

\begin{align*}
&   \sup_{\| t \|_{\infty} \leq T_n } \left|  \varphi_{\overline{F}_0}(t) - \varphi_{F'}(t)  \right| \leq   T_n^{-1} n^{-C_1} \epsilon_n   \\ &  \sup_{\| t \|_{\infty} \leq T_n} \|  \nabla \varphi_{\overline{F}_0} (t) - \nabla \varphi_{F'}(t)   \| \leq n^{-C_1} \epsilon_n.
\end{align*}

From the final claim of Lemma <ref>, we can also assume without loss of generality that the support points satisfy $  \inf_{k \neq j} \| \mu_k - \mu_j   \| \geq \epsilon_n^{L_2}  $ for some $L_2 > 1$. Observe that $2 \inf_{\| t \|_{\infty} \leq T_n }  \left| \varphi_{\overline{F}_0}(t)  \right|  \geq  \inf_{\| t \|_{\infty} \leq T_n }  \left| \varphi_{F_0}(t) \right| \geq n^{-C_1}$ and

\begin{align*}
\sup_{\| t \|_{\infty} \leq T_n }  \| \nabla \log \varphi_{ \overline{F}_0}  (t)  \|  & \leq \sup_{\| t \|_{\infty} \leq T_n}  \| \nabla \log  \varphi_{F_0}  (t) - \nabla \log \varphi_{\overline{F}_0}(t)     \| +  \sup_{\| t \|_{\infty} \leq T_n }  \| \nabla \log \varphi_{ F_0 }  (t)  \| \\ & \leq D \big( \epsilon_n + T_n \big) \\ & \leq D T_n.
\end{align*}

The preceding bounds imply
\begin{align*}
&  \sup_{\| t \|_{\infty} \leq T_n}  \|  \nabla \log  \varphi_{ \overline{F}_0  } (t)  \| \frac{ \left|  \varphi_{\overline{F}_0}(t) - \varphi_{F'}(t)  \right|}{ \left|  \varphi_{\overline{F}_0}(t)  \right|} \leq  D \epsilon_n \; , \; \sup_{\| t \|_{\infty} \leq T_n} \frac{ \|  \nabla \varphi_{\overline{F}_0} (t) - \nabla \varphi_{F'}(t)   \|}{\left| \varphi_{\overline{F}_0}(t)  \right|} \leq D \epsilon_n.
\end{align*}

From an analogous argument to the bound for $F_0$ and $\overline{F}_0$, it follows that

\begin{align*}
\sup_{\| t \|_{\infty} \leq T_n}  \| \nabla \log  \varphi_{ \overline{F}_0 }  (t) - \nabla \log \varphi_{F'}(t)     \| \leq D \epsilon_n.
\end{align*}

Fix any $L_3 > L_2$  sufficiently large such that
\begin{align*} T_n^2 n^{C_1} \epsilon_n^{L_3} \leq \epsilon_n  \; \; , \; \; n^{C_1} \sqrt{n}  \epsilon_n \sqrt{\log n} \epsilon_n^{L_3/2}  \leq \epsilon_n \; , \;   T_n n^{C_1} (\log \epsilon_n^{-1})^{1/\chi} \epsilon_n^{L_3} \leq \epsilon_n.
\end{align*}

Define $V_i = \{ t \in I_{n} :  \| t - \mu_i  \| \leq  \epsilon_n^{L_3}  \} $ for $i=1,\dots,N$ and set $V_0 = \R^d \setminus \bigcup_{i=1}^N V_i$. From the definition of the $\{\mu_i \}_{i=1}^N$, it follows that $\{V_0, V_1, \dots , V_N  \}$ is a disjoint partition of $\R^d$. By Lemma <ref>, for any distribution $P$ that satisfies $  \int_{\R^d} \| x \|^2 dP(x) \leq n \epsilon_n^2 \log n  $ and $ \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3} $, we have that
\begin{align*}
&  \sup_{\| t \|_{\infty} \leq T_n}  \|  \nabla  \log \varphi_{F'}(t)  \|  \frac{ \left|  \varphi_{P}(t) - \varphi_{F'}(t)  \right|}{\left|  \varphi_{F'}(t)  \right|} \leq  D \epsilon_n \; , \; \sup_{\| t \|_{\infty} \leq T_n} \frac{ \|  \nabla \varphi_{P} (t) - \nabla \varphi_{F'}(t)   \|}{\left| \varphi_{F'}(t)  \right|} \leq D \epsilon_n.
\end{align*}

For all such $P$, an analogous argument to the bound for $F_0$ and $\overline{F}_0$ implies that
\begin{align*}
\sup_{\| t \|_{\infty} \leq T_n}  \| \nabla \log  \varphi_{ P }  (t) - \nabla \log \varphi_{F'}(t)     \| \leq D \epsilon_n.
\end{align*}

From combining all the preceding bounds, observe that such all such $P$  also satisfy
\begin{align*} &
\sup_{\| t \|_{\infty} \leq T_n}  \| \nabla \log  \varphi_{ P }  (t) - \nabla \log \varphi_{F_0}(t)     \| \leq D \epsilon_n
\\ &
\sup_{\| t \|_{\infty} \leq T_n } \|  \nabla \log \varphi_P(t)   \|   \leq D \epsilon_n + \sup_{\| t \|_{\infty} \leq T_n } \|  \nabla \log \varphi_{F_0}(t)  \| \leq D T_n.  \end{align*}

Given any positive definite $\Sigma$, the preceding bound also implies that
\begin{align*}
\sup_{\| t \|_{\infty} \leq T_n } \|  \nabla \log \varphi_{P,\Sigma}(t)   & \| \leq \sup_{\| t \|_{\infty} \leq T_n } \bigg[ \|  \nabla \log \varphi_{P}(t)   \| + \|  \nabla \log \varphi_{\Sigma}(t)   \|   \bigg] \\ & \leq D  \bigg( T_n + T_n \| \Sigma \| \bigg).
\end{align*}

By Lemma <ref> and <ref>, we have
\begin{align*}
&    \int_{\mathbb{B}(T_n)}   \left|  \widehat{\varphi}_{Y_2}(t) - \varphi_{Y_2}(t)    \right|^2 dt \leq D \frac{ T_n^d  \log(T_n)  }{n}  \; ,  \\ &  \int_{\mathbb{B}(T_n)}   \|  \widehat{\varphi}_{Y_1,Y_2}(t) - \varphi_{Y_1,Y_2}(t)   \|^2 dt \leq D \frac{  T_n^d \log(T_n)  }{n} .
\end{align*}

with $\mathbb{P}$ probability approaching $1$. On this set, when $P$ satisfies the conditions specified above and $\| \Sigma \| \leq D$, we have that

\begin{align*}
\|  \widehat{\varphi}_{Y_2} \nabla \log \varphi_{P,\Sigma} - \widehat{\varphi}_{Y_1,Y_2}   \|_{\mathbb{B}(T_n)}^2 & \leq D \bigg[  \|  {\varphi}_{Y_2} \nabla \log \varphi_{P,\Sigma} -{\varphi}_{Y_1,Y_2}   \|_{\mathbb{B}(T_n)}^2  +  \frac{T_n^{d+1} \log(T_n) }{n}  \bigg] \\ & \leq D \bigg[  \|  {\varphi}_{Y_2} \nabla \log \varphi_{P,\Sigma} -{\varphi}_{Y_1,Y_2}   \|_{\mathbb{B}(T_n)}^2  +  \epsilon_n^2 \bigg].
\end{align*}

Define the set
$$ \mathcal{G}_n = \bigg \{ (P,\Sigma) : \|  \Sigma - \Sigma_0 \| \leq \epsilon_n^2 , \int_{\R^d} \| x \|^2 dP(x) \leq n \epsilon_n^2 \log(n) \: ,   \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3}   \bigg  \}.  $$

From combining all the preceding bounds, it follows that
\begin{align*}
&   \int  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \\ & \geq  \int_{ \mathcal{G}_n }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \exp \big(  -nD \epsilon_n^2 \big) \int_{ \mathcal{G}_n }  \exp \bigg( - nD    \|  {\varphi}_{Y_2} \nabla \log \varphi_{P,\Sigma} -{\varphi}_{Y_1,Y_2}   \|_{\mathbb{B}(T_n)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma).
\end{align*}

Since $ \varphi_{Y_1,Y_2} = \varphi_{Y_2} \nabla \log \varphi_{X}  $ and $ \varphi_{X} = \varphi_{F_0,\Sigma_0} $, the preceding integral can be expressed as
\begin{align*}
&  \int_{ \mathcal{G}_n }  \exp \bigg( - nD    \|  {\varphi}_{Y_2} \nabla \log \varphi_{F,\Sigma} -{\varphi}_{Y_1,Y_2}   \|_{\mathbb{B}(T_n)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \\ & = \int_{ \mathcal{G}_n }  \exp \bigg( - nD    \|  {\varphi}_{Y_2} \big( \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{F_0,\Sigma_0}     \big)  \|_{\mathbb{B}(T_n)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma).
\end{align*}

For every $(P,\Sigma) \in \mathcal{G}_n$, the preceding bounds imply that
\begin{align*}
&  \sup_{\|t \|_{\infty} \leq T_n } \|  \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{F_0,\Sigma_0}    \| \\ &  = \sup_{\|t \|_{\infty} \leq T_n } \|   \big (\nabla \log \varphi_{P} - \nabla \log \varphi_{F_0} \big) + \big(  \nabla \log \varphi_{\Sigma} -\nabla \log \varphi_{\Sigma_0}    \big)  \| \\ & \leq \sup_{\|t \|_{\infty}\leq T_n } \|   \nabla \log \varphi_{P} - \nabla \log \varphi_{F_0}  \| + \sup_{\|t \|_{\infty}\leq T_n } \|   \nabla \log \varphi_{\Sigma} - \nabla \log \varphi_{\Sigma_0}  \| \\ & \leq D \epsilon_n + D \| \Sigma - \Sigma_0 \| T_n \\ & \leq D \epsilon_n \: ,
\end{align*}

where we used that $ \| \Sigma - \Sigma_0  \| \leq \epsilon_n^2$ and $\epsilon_n^2 T_n \leq \epsilon_n$. Since $ \| \varphi_{Y_2}  \|_{L^2} < \infty$, it follows that
\begin{align*}
& \int_{ \mathcal{G}_n }  \exp \bigg( - nD    \|  {\varphi}_{Y_2} \big( \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{F_0,\Sigma_0}     \big)  \|_{\mathbb{B}(T_n)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \exp \big(  -nD \epsilon_n^2 \big) \int_{ \mathcal{G}_n }  d \nu_{\alpha,G} (P,\Sigma).
\end{align*}

Define the sets
\begin{align*}
&  \mathcal{G}_{n,1} = \bigg \{ (P,\Sigma) : \|  \Sigma - \Sigma_0 \| \leq \epsilon_n^2  \: ,   \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3}   \bigg  \} \; , \\ &  \mathcal{G}_{n,2} = \bigg \{ (P,\Sigma) : \|  \Sigma - \Sigma_0 \| \leq \epsilon_n^2 , \int_{\R^d} \| x \|^2 dP(x) > n \epsilon_n^2 \log(n) \: ,   \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3}   \bigg  \} .
\end{align*}

Observe that $\mathcal{G}_n = \mathcal{G}_{n,1} \setminus \mathcal{G}_{n,2} $ and hence
\begin{align}
\int_{ \mathcal{G}_n }  d \nu_{\alpha,G} (P,\Sigma) = \int_{ \mathcal{G}_{n,1} }  d \nu_{\alpha,G} (P,\Sigma)  - \int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G}  (P,\Sigma) .    \label{setdiff}
\end{align}

For the second term in (<ref>), we have
\begin{align*}
\int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G} (P,\Sigma) &  \leq \int_{ P : \int_{\R^d} \| x \|^2 dP(x) > n \epsilon_n^2 \log(n)  }  d \nu_{\alpha,G} (P,\Sigma) \\ & \leq  \int_{ P : \int_{\R^d} \| x \|^2 dP(x) > n \epsilon_n^2 \log(n)  }  d \text{DP}_{\alpha}(P) \\ &  \leq \exp(-D n \epsilon_n^2 \log n) \; ,
\end{align*}

where the second inequality is due to $\nu_{\alpha,G}$ being a product measure $\nu_{\alpha,G} = \text{DP}_{\alpha} \otimes G$ and the third inequality follows from an application of Lemma <ref>.

For the first term in (<ref>), we have that
\begin{align*}
\int_{\mathcal{G}_{n,1}} d \nu_{\alpha,G} (P,\Sigma) = \int_{ \Sigma : \| \Sigma - \Sigma_0  \| \leq  \epsilon_n^2} \int_{P : \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3}    } d  \text{DP}_{\alpha}(P)  d G(\Sigma).
\end{align*}

As $ \text{DP}_{\alpha} $ is constructed using a Gaussian base measure $\alpha $, it is straightforward to verify that $
\inf_{j=1}^N \alpha(V_j) \geq C \epsilon_n^{L_3 d} \exp(- C' ( \log \epsilon_n^{-1})^{2/\chi}  ) $ for universal constants $C,C' > 0$. By definition of $ \text{DP}_{\alpha}$, $(P(V_1),\dots, P(V_N)) \sim \text{Dir}(N, \alpha(V_1), \dots,  \alpha(V_N))$. As $N = D \{ \log ( \epsilon_n^{-1} )  \}^{\iota}$, an application of <cit.> implies
\begin{align*}
\int_{P : \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n    } d  \text{DP}_{\alpha}(P) \geq C \exp \big (  - C'  ( \log \epsilon_n^{-1} )^{\iota + \max \{2/ \chi,1   \}  }   \big)  & = C \exp \big (  - C'  ( \log \epsilon_n^{-1} )^{\lambda  }   \big) \\ & \geq  C \exp \big (  - C''  n \epsilon_n^2   \big) .
\end{align*}

It remains to bound the outer integral. By Assumption <ref>, there exists a universal constant $C > 0$  and $q > 0$ such that

\begin{align}
\label{covlb}     \int_{ \Sigma : \| \Sigma - \Sigma_0  \| \leq  \epsilon_n^2} d G(\Sigma)  \geq C \epsilon_n^q.
\end{align}

From combining the preceding bounds, it follows from (<ref>) that
\begin{align*}
\int_{ \mathcal{G}_n }  d \nu_{\alpha,G} (P,\Sigma) & = \int_{ \mathcal{G}_{n,1} }  d \nu_{\alpha,G} (P,\Sigma)  - \int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G} (P,\Sigma)  \\ & \geq C \exp(- C'' n \epsilon_n^2) - \exp(-D n \epsilon_n^2 \log n) \\ & \geq \exp(-Dn \epsilon_n^2).
\end{align*}

The  estimate for the lower bound of the normalizing constant follows from combining all the preceding bounds.

We prove the main statement of the theorem. Observe that for any $E > 0$, we have

\begin{align*}
&  \int \limits_{ (P,\Sigma) : \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2 > 2E \epsilon_n^2 }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ &  \leq \exp \big( - n E \epsilon_n^2    \big).
\end{align*}

From combining the preceding bound with the one derived in step $(i)$, it follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2 \leq M^2 \epsilon_n^2 \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\geq 1- \exp(-C' n \epsilon_n^2).
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

By an application of Lemma <ref>, we have that
\begin{align*}
\|  \widehat{\varphi}_{Y_1,Y_2}  - \varphi_{Y_1,Y_2}   \|_{\mathbb{B}(T_n)} \leq D \epsilon_n
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

Since $\varphi_{Y_1,Y_2} = \varphi_{Y_2} \nabla \log  \varphi_{F_0,\Sigma_0} $ and  $  \sup_{\| t \|_{\infty} \leq T_n} \| \nabla \log  \varphi_{F_0,\Sigma_0} \| \leq D T_n $, an application of Lemma <ref> implies that
\begin{align*}
\|   \widehat{\varphi}_{Y_2}  \nabla \log \varphi_{F_0,\Sigma_0}  - \varphi_{Y_1,Y_2}  \|_{\mathbb{B}(T_n)} \leq D \epsilon_n
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

From combining the preceding bounds, it follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \| \widehat{\varphi}_{Y_2} \big( \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{F_0,\Sigma_0} \big)         \|_{ \mathbb{B}(T_n) }^2 \leq M^2  \epsilon_n^2 \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2)
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

Since $T_n =  (c_0 \log n)^{1/ 2} $ for some $c_0$ satisfying $c_0 R  = \gamma < 1/2$, an application of Lemma <ref> implies that
\begin{align*}
\sup_{\| t \|_{\infty} \leq T_n}   \frac{  \left| \widehat{\varphi}_{Y_2}(t)  - \varphi_{Y_2}(t)    \right| }{\left| \varphi_{Y_2}(t) \right|} \leq D n^{-1/2 + \gamma} \sqrt{\log \log  n}
\end{align*}

with $\mathbb{P}$ probability approaching $1$. As the quantity on the right converges to zero, it follows that

\begin{align*}
\left|   \widehat{\varphi}_{Y_2}(t)    \right|  & \geq  \left|   {\varphi}_{Y_2}(t)    \right| -  \left| \widehat{\varphi}_{Y_2}(t)  - \varphi_{Y_2}(t)    \right|  \geq \frac{1}{2} \left|   {\varphi}_{Y_2}(t)   \right| \geq \frac{1}{2} n^{-\gamma}
\end{align*}

uniformly over the set $ \{ \| t \|_{\infty} \leq T_n \}$, with $\mathbb{P}$ probability approaching $1$. It follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \|  \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{F_0,\Sigma_0}          \|_{ \mathbb{B}(T_n) }^2 \leq M^2  n^{2 \gamma} \epsilon_n^2 \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2)
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$. The claim follows from observing that
$$ \epsilon_n n^{\gamma} \asymp n^{-1/2+\gamma} (\log n)^{\lambda/2} . $$

We use $D > 0$ as a generic universal constant that may change from line to line.
\begin{align*}
\epsilon_n^2 = n^{-1} (\log n)^{\lambda} \; \; \; \; ,  \; \; \; \;  \lambda = \begin{cases}
\frac{3}{\chi}  +  \frac{1}{2}  & \chi < 2 \\ 2 & \chi \geq 2.
\end{cases}
\end{align*}

The proof continues from the conclusion of Theorem <ref> with a few modifications to the preceding bounds to account for the change in covariance prior. For ease of notation, we suppress the dependence of $G = G_n$ on $n$.

First, we derive a lower bound for the normalizing constant of the posterior measure.  Specifically, we aim to show that there exists a $C > 0 $ such that
\begin{align}
\label{t1lb}  \int  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \geq \exp \big(   - C n    \epsilon_{n}^2   \big)
\end{align}

holds with $\mathbb{P}$ probability approaching $1$.

The argument is identical to part $(i)$ of Theorem $\ref{deconv-prod-t1}$ except that Equation $(\ref{covlb})$ is replaced with the following argument. The law of $G$ is given by $\Omega / \sigma_n^2$ where $\Omega \sim L$ and $L$ is a probability measure on $\mathbf{S}_+^d$ that  satisfies Assumption <ref>. By Assumption <ref> and the definition of $(\sigma_n^2, \epsilon_n^2)$, there exists universal constant $C,C',C'' > 0$ such that
\begin{align*}
\int_{ \Sigma : \| \Sigma - \Sigma_0  \| \leq  \epsilon_n^2} d G(\Sigma) = \int_{\Sigma : \| \Sigma - \sigma_n^2 \Sigma_0   \| \leq \sigma_n^2 \epsilon_n^2  } d L(\Sigma) \geq C  \exp \big( - C' \sigma_n^{-2 \kappa}   \big) \geq C \exp ( -C'' n \epsilon_n^2  ).
\end{align*}

The lower bound for the normalizing constant then follows from the conclusion of part $(i)$ of Theorem $\ref{deconv-prod-t1}$.

Next, we establish a preliminary local concentration bound under the prior. Observe that for any $E > 0$, we have

\begin{align*}
&  \int \limits_{ (P,\Sigma) : \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2 > 2E \epsilon_n^2 }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2        \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ &  \leq \exp \big( - n E \epsilon_n^2    \big).
\end{align*}

The law of $G$ is given by $\Sigma / \sigma_n^2$ where $\Sigma \sim L$ and $L$ is a probability measure on  $\mathbf{S}_+^d$ that  satisfies Assumption <ref>. By Assumption <ref>, it follows that for every $E' > 0$, there exists $E > 0$ such that

\begin{align*}
\int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\Sigma) = \int_{\Sigma : \| \Sigma^{-1}  \| > E (n \epsilon_n^2)^{1/ \kappa}  }   d L(\Sigma) \leq \exp \big(  - E' n \epsilon_n^2  \big).
\end{align*}

As the prior is a product measure $\nu_{\alpha,G} = \text{DP}_{\alpha} \otimes G$ and  $ \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2 \geq 0$, the preceding bound implies
\begin{align*}
&  \int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2        \bigg) d \nu_{\alpha,G} (P,\Sigma)  \\ & \leq  \int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\Sigma)  \\ & \leq \exp \big(  - E' n \epsilon_n^2  \big).
\end{align*}

From combining the preceding bounds, it follows that for any $E' > 0$ we can pick $E > 0$ sufficiently large such that
\begin{align*} \int \limits_{\substack{(P,\Sigma) : \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2 \leq E \epsilon_n^2 , \\ \| \Sigma^{-1} \| \leq E \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}}}
\exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )      & - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2        \bigg)    d \nu_{\alpha,G} (P,\Sigma) \\ & \geq 1- \exp \big( - E' n \epsilon_n^2   \big) .
\end{align*}

We prove the main statement of the theorem. From the bounds derived in steps $(i)$ and $(ii)$, it follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{ \mathbb{B}(T_n) }^2 \leq M^2 \epsilon_n^2, \| \Sigma^{-1} \| \leq M^2 \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2)
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

By an application of Lemma <ref>, we have that
\begin{align*}
\|  \widehat{\varphi}_{Y_1,Y_2}  - \varphi_{Y_1,Y_2}   \|_{\mathbb{B}(T_n)} \leq D \epsilon_n
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

Since $\varphi_{Y_1,Y_2} = \varphi_{Y_2} \nabla \log  \varphi_{F_0,\Sigma_0} $ and  $  \sup_{\| t \|_{\infty} \leq T_n} \| \nabla \log  \varphi_{F_0,\Sigma_0} \| \leq D T_n $, an application of Lemma <ref> implies that
\begin{align*}
\|   \widehat{\varphi}_{Y_2}  \nabla \log \varphi_{F_0,\Sigma_0}  - \varphi_{Y_1,Y_2}  \|_{\mathbb{B}(T_n)} \leq D  \epsilon_n
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

From combining the preceding bounds, it follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \| \widehat{\varphi}_{Y_2} \big( \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{F_0,\Sigma_0} \big)         \|_{ \mathbb{B}(T_n) }^2 \leq M^2  \epsilon_n^2, \| \Sigma^{-1} \| \leq M^2 \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2)
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

Since $T_n =  (c_0 \log n)^{1/ 2} $ for some $c_0$ satisfying $c_0 R  = \gamma < 1/2$, an application of Lemma <ref> implies that
\begin{align*}
\sup_{\| t \|_{\infty} \leq T_n}   \frac{  \left| \widehat{\varphi}_{Y_2}(t)  - \varphi_{Y_2}(t)    \right| }{\left| \varphi_{Y_2}(t) \right|} \leq D n^{-1/2 + \gamma} \sqrt{\log \log  n}
\end{align*}

with $\mathbb{P}$ probability approaching $1$. As the quantity on the right converges to zero, it follows that

\begin{align*}
\left|   \widehat{\varphi}_{Y_2}(t)    \right|  & \geq  \left|   {\varphi}_{Y_2}(t)    \right| -  \left| \widehat{\varphi}_{Y_2}(t)  - \varphi_{Y_2}(t)    \right|  \geq \frac{1}{2} \left|   {\varphi}_{Y_2}(t)   \right| \geq \frac{1}{2} n^{-\gamma}.
\end{align*}

uniformly over the set $ \{ \| t \|_{\infty} \leq T_n \}$, with $\mathbb{P}$ probability approaching $1$. It follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \|  \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{F_0,\Sigma_0}         \|_{ \mathbb{B}(T_n) }^2 \leq M^2 n^{2 \gamma} \epsilon_n^2, \| \Sigma^{-1} \| \leq M^2  \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2).
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

In dimension $d=1$, the fundemental theorem of calculus, Cauchy-Schwarz and the initial value condition $ \nabla \log \varphi_X (0) = \nabla \log \varphi_{P,\Sigma}(0) = 0 $  imply that
\begin{align*}
\left|\log \varphi_{F_0,\Sigma_0} (t) - \log \varphi_{P,\Sigma}(t)           \right| & = \left| \int_0^t  \big[  \nabla \log \varphi_{F_0,\Sigma_0} (s) - \nabla \log \varphi_{P,\Sigma}(s)     ]  ds  \right| \\ & \leq \sqrt{T_n} \| \nabla \log  \varphi_{F_0,\Sigma_0}  - \nabla \log \varphi_{P,\Sigma}      \|_{\mathbb{B}(T_n)}
\end{align*}

holds for every $t \in \mathbb{B}(T_n)$. Furthermore, for every fixed $t \in \R^d$, the mean value theorem implies that
\begin{align*}
\left|  \varphi_{F_0,\Sigma_0}(t) -   \varphi_{P,\Sigma}(t)   \right|  \leq \sup_{s_t \in [0,1]}  \left| e^{s_t  \log  \varphi_{F_0,\Sigma_0}(t) + (1-s_t) \log \varphi_{P,\Sigma}(t)  } \right| \left|  \log \varphi_{F_0,\Sigma_0} (t) - \log \varphi_{P,\Sigma}(t)    \right|.
\end{align*}

Since $ \left| \varphi_{F_0,\Sigma_0} \right| \leq 1 $ and $\left| \varphi_{P,\Sigma}   \right| \leq 1$ (as they are characteristic function of random variables), the preceding bound reduces to
\begin{align*}
\left|  \varphi_{F_0,\Sigma_0}(t) -   \varphi_{P,\Sigma}(t)   \right|  \leq  \left|  \log \varphi_{F_0,\Sigma_0} (t) - \log \varphi_{P,\Sigma}(t)    \right|.
\end{align*}

From combining the preceding bounds and noting that the Lebesgue measure of $\mathbb{B}(T_n)$ is of order $T_n$, it follows that there exists a universal constant $D > 0$ such that
\begin{align*}
\|   \varphi_{F_0,\Sigma_0} -  \varphi_{P,\Sigma}  \|_{\mathbb{B}(T_n)} \leq D T_n \| \nabla \log  \varphi_{F_0,\Sigma_0}  - \nabla \log \varphi_{P,\Sigma}      \|_{\mathbb{B}(T_n)} .
\end{align*}

It follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \|   \varphi_{P,\Sigma} -  \varphi_{F_0,\Sigma_0}         \|_{ \mathbb{B}(T_n) } \leq M T_n n^{ \gamma} \epsilon_n, \| \Sigma^{-1} \| \leq M^2  \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2).
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

It remains to examine the bias from truncating the $L^2$ norm to the set $\mathbb{B}(T_n)$. Suppose $ \|  \Sigma^{-1}  \| \leq M^2  \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}  $ holds. It follows that there exists a $c > 0$ for which $  \lambda_1(\Sigma ) \geq c (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} $ holds. From the definition of $\sigma_n^2$, we have  $T_n^2 (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} \asymp  \log n$. It follows that there exists a universal constant $C > 0$ such that

\begin{align*}
\| \big( {\varphi}_{F_0,\Sigma_0}   -   {\varphi}_{P,\Sigma} \big) \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 & \leq  2 \|  {\varphi}_{F_0,\Sigma_0}    \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 +  2 \|     {\varphi}_{P,\Sigma}  \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 \\ & \leq 2 \int_{\| t \|_{\infty} > T_n}  e^{- t' \Sigma_0 t} dt + 2 \int_{\| t \|_{\infty} > T_n} e^{- t' \Sigma t} dt \\ & \leq 2 \int_{\| t \|_{\infty} > T_n}  e^{- t' \Sigma_0 t} dt + 2 \int_{\| t \|_{\infty} > T_n} e^{- c \| t \|^2  \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa}   } dt \\ & \leq D \bigg[  e^{- C T_n^2} T_n^{d-2} +  \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa}   e^{- C T_n^2 \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa} } T_n^{d-2}   \bigg ] .
\end{align*}

From substituting $T_n^2 (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} \asymp  \log n$, the preceding bound reduces to $D n^{- 2 K  }$ for some constant $ K \in (0,1/2]$ (that depends on, among other factors, the smallest eigenvalue of $\Sigma_0$).

With constant $K$ as specified above, it follows that for any $C' > 0$, there exists $ M > 0$ such that
$$ \nu_{\alpha,G} \bigg(  \| \varphi_{F_0,\Sigma_0}  -  \varphi_{P,\Sigma}      \|_{L^2} \leq M T_n n^{\gamma} \epsilon_n  + n^{-K}  \:     \big|\mathcal{Z}_n,T_n  \bigg) \geq 1- \exp(-C' n \epsilon_n^2)  $$

holds with $\mathbb{P}$ probability approaching $1$. The claim follows from observing that
$$ T_n n^{\gamma} \epsilon_n \asymp n^{-1/2 + \gamma} (\log n)^{\lambda/2 + 1/2}.  $$

The proof proceeds through several steps which we outline below. We use $D > 0$ as a generic universal constant that may change from line to line.
\begin{align*}
\epsilon_n^2 = n^{-1} (\log n)^{\lambda} \; \; \; \; ,  \; \; \; \;  \lambda = \begin{cases}
\chi^{-1}(d +2) + d/2 & \chi < 2 \\ d + 1 & \chi \geq 2.
\end{cases}
\end{align*}

For ease of notation, we supress the dependence of $G = G_n$ on $n$.

First, we derive a lower bound for the normalizing constant of the posterior measure.  Specifically, we aim to show that there exists a $C > 0 $ such that
\begin{align}
\label{t1lb}  \int  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \geq \exp \big(   - C n    \epsilon_{n}^2   \big)
\end{align}

holds with $\mathbb{P}$ probability approaching $1$.

The argument to verify this is completely analogous to part $(i)$ of Theorem <ref>. To be specific, all the estimates in that proof hold uniformly over $\| t \|_{\infty} \leq T_n$ and so they also hold over the set $  \{  tz : \| z \|_{\infty} \leq T_n \; , \; t \in [0,1]  \}$. Furthermore, as $\mathcal{H}^{d-1}( \partial \mathbb{B}(T_n)) \leq D T_n^{d-1}$, Lemma <ref> and <ref> imply that
\begin{align*}
&    \int_{\partial \mathbb{B}(T_n)}   \left|  \widehat{\varphi}_{Y_2}(t) - \varphi_{Y_2}(t)    \right|^2 d \mathcal{H}^{d-1}(t) \leq D \frac{ T_n^{d-1}  \log(T_n)  }{n}  \; ,  \\ &  \int_{ \partial \mathbb{B}(T_n)}   \|  \widehat{\varphi}_{Y_1,Y_2}(t) - \varphi_{Y_1,Y_2}(t)   \|^2 d \mathcal{H}^{d-1}(t) \leq D \frac{  T_n^{d-1} \log(T_n)  }{n}
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$. The only other significant change is that Equation $(\ref{covlb})$ is replaced with the following argument. The law of $G$ is given by $\Omega / \sigma_n^2$ where $\Omega \sim L$ and $L$ is a probability measure on  $\mathbf{S}_+^d$ that  satisfies Assumption <ref>. By Assumption <ref> and the definition of $\sigma_n^2$, there exists a universal constant $C,C',C'' > 0$ such that
\begin{align*}
\int_{ \Sigma : \| \Sigma - \Sigma_0  \| \leq  \epsilon_n^2} d G(\Sigma) = \int_{\Sigma : \| \Sigma - \sigma_n^2 \Sigma_0   \| \leq \sigma_n^2 \epsilon_n^2  } d L(\Sigma) \geq C  \exp \big( - C' \sigma_n^{-2 \kappa}   \big) \geq C \exp ( -C'' n \epsilon_n^2  ).
\end{align*}

Next, we establish a preliminary local concentration bound under the prior. Observe that for any $E > 0$, we have

\begin{align*}
&  \int \limits_{ (P,\Sigma) : \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2     > 2E \epsilon_n^2 }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2           \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ &  \leq \exp \big( - n E \epsilon_n^2    \big).
\end{align*}

The law of $G$ is given by $\Sigma / \sigma_n^2$ where $\Sigma \sim L$ and $L$ is a probability measure on  $\mathbf{S}_+^d$ that  satisfies Assumption <ref>. By Assumption <ref>, it follows that for every $E' > 0$, there exists $E > 0$ such that

\begin{align*}
\int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\Sigma) = \int_{\Sigma : \| \Sigma^{-1}  \| > E (n \epsilon_n^2)^{1/ \kappa}  }   d L(\Sigma) \leq \exp \big(  - E' n \epsilon_n^2  \big).
\end{align*}

As the prior is a product measure $\nu_{\alpha,G} = \text{DP}_{\alpha} \otimes G$ and  $ \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2  \geq 0$, the preceding bound implies
\begin{align*}
&  \int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2            \bigg) d \nu_{\alpha,G} (P,\Sigma)  \\ & \leq  \int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\Sigma)  \\ & \leq \exp \big(  - E' n \epsilon_n^2  \big).
\end{align*}

From combining the preceding bounds, it follows that for any $E' > 0$ we can pick $E > 0$ sufficiently large such that
\begin{align*} \int \limits_{\substack{(P,\Sigma) : \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2    \leq E \epsilon_n^2 , \\ \| \Sigma^{-1} \| \leq E \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}}}
\exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )      & - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2            \bigg)    d \nu_{\alpha,G} (P,\Sigma) \\ & \geq 1- \exp \big( - E' n \epsilon_n^2   \big) .
\end{align*}

We prove the main statement of the theorem. From the bounds derived in steps $(i)$ and $(ii)$, it follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2    \leq M^2 \epsilon_n^2, \| \Sigma^{-1} \| \leq M^2 \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2)
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

By an application of Lemma <ref>, we have that
\begin{align*}
\|  \widehat{\varphi}_{Y_1,Y_2}  - \varphi_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}    \leq D \epsilon_n
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

Since $\varphi_{Y_1,Y_2} = \varphi_{Y_2} \nabla \log  \varphi_{F_0,\Sigma_0} $ and  $  \sup_{\| t \|_{\infty} \leq T_n} \| \nabla \log  \varphi_{F_0,\Sigma_0} \| \leq D T_n $, an application of Lemma <ref> implies that
\begin{align*}
\|   \widehat{\varphi}_{Y_2}  \nabla \log \varphi_{F_0,\Sigma_0}  - \varphi_{Y_1,Y_2}  \|_{\partial,\;\mathbb{B}(T)}    \leq D  \epsilon_n
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

From combining the preceding bounds, it follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \| \widehat{\varphi}_{Y_2} \big( \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{F_0,\Sigma_0} \big)         \|_{\partial,\;\mathbb{B}(T)}^2     \leq M^2  \epsilon_n^2, \| \Sigma^{-1} \| \leq M^2 \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2)
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

Since $T_n =  (c_0 \log n)^{1/ 2} $ for some $c_0$ satisfying $c_0 R  = \gamma < 1/2$, an application of Lemma <ref> implies that
\begin{align*}
\sup_{\| t \|_{\infty} \leq T_n}   \frac{  \left| \widehat{\varphi}_{Y_2}(t)  - \varphi_{Y_2}(t)    \right| }{\left| \varphi_{Y_2}(t) \right|} \leq D n^{-1/2 + \gamma} \sqrt{\log \log  n}
\end{align*}

with $\mathbb{P}$ probability approaching $1$. As the quantity on the right converges to zero, it follows that

\begin{align*}
\left|   \widehat{\varphi}_{Y_2}(t)    \right|  & \geq  \left|   {\varphi}_{Y_2}(t)    \right| -  \left| \widehat{\varphi}_{Y_2}(t)  - \varphi_{Y_2}(t)    \right|  \geq \frac{1}{2} \left|   {\varphi}_{Y_2}(t)   \right| \geq \frac{1}{2} n^{-\gamma}.
\end{align*}

uniformly over the set $ \{ \| t \|_{\infty} \leq T_n \}$, with $\mathbb{P}$ probability approaching $1$. It follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \|  \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{F_0,\Sigma_0}         \|_{\partial,\;\mathbb{B}(T)}^2     \leq M^2 n^{2 \gamma} \epsilon_n^2, \| \Sigma^{-1} \| \leq M^2  \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2).
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

As $\varphi_{P,\Sigma}$ and $\varphi_{F_0,\Sigma_0}$ are characteristic functions, they satisfy the initial value condition $  \log \varphi_{P,\Sigma} (0) = \log \varphi_{F_0,\Sigma_0}(0) = 0 $. In particular, every $z \in \partial \mathbb{B}(T_n)$ can be expressed as
\begin{align*}
\log \varphi_{P,\Sigma} (z) - \log \varphi_{F_0,\Sigma_0}(z) = \int_{0}^1 \langle \nabla \log \varphi_{P,\Sigma} (tz) - \nabla \log \varphi_{F_0,\Sigma_0}(tz)  , z    \rangle dt .
\end{align*}

By Cauchy-Schwarz and Jensen's inequality, this implies that
\begin{align*}
\left|   \log \varphi_{P,\Sigma} (z) - \log \varphi_{F_0,\Sigma_0}(z)  \right|^2 \leq  \| z \|^2 \int_{0}^1 \| \nabla \log \varphi_{P,\Sigma}(tz) - \nabla \log \varphi_{F_0,\Sigma_0} (tz) \|^2 dt.
\end{align*}

In particular since $\| z \| \leq D T_n$ for every $z \in \partial \mathbb{B}(T_n)$, we obtain the bound
\begin{align*}
& \int_{\partial \mathbb{B}(T_n) }  \left| \log \varphi_{P,\Sigma} (z) - \log \varphi_{F_0,\Sigma_0}(z)  \right|^2 d \mathcal{H}^{d-1}(z) \\ &  \leq T_n^2 \int_{\partial \mathbb{B}(T_n)} \int_{0}^1   \| \nabla \log \varphi_{P,\Sigma}(tz) - \nabla \log \varphi_{F_0,\Sigma_0} (tz) \|^2 dt d \mathcal{H}^{d-1}(z) .
\end{align*}

Suppose $(P,\Sigma)$ is such that  $\|  \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{F_0,\Sigma_0}         \|_{\partial,\;\mathbb{B}(T)}^2     \leq M^2 n^{2 \gamma} \epsilon_n^2$. By definition of the metric, this means that
\begin{align*}
& \int_{\mathbb{B}(T_n) } \| \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{F_0,\Sigma_0}    \|^2 dt  \leq M^2 n^{2 \gamma} \epsilon_n^2 \; , \\ & \int_{\partial \mathbb{B}(T_n)} \int_{0}^1   \| \nabla \log \varphi_{P,\Sigma}(tz) - \nabla \log \varphi_{F_0,\Sigma_0} (tz) \|^2 dt d \mathcal{H}^{d-1}(z) \leq M^2 n^{2 \gamma} \epsilon_n^2.
\end{align*}

From the preceding bounds and the Poincaré inequality (Lemma <ref>), it follows that there exists a universal constant $D > 0$ such that
\begin{align*}
\|   \log \varphi_{P,\Sigma}  - \log \varphi_{F_0,\Sigma_0}  \|_{\mathbb{B}(T_n)}^2  \leq D M^2 T_n^3 n^{2 \gamma} \epsilon_n^2.
\end{align*}

Since $\left|  \varphi_{X}(t) -   \varphi_{P,\Sigma}(t)   \right|  \leq  \left|  \log \varphi_X (t) - \log \varphi_{P,\Sigma}(t)    \right|$ for every $t \in \R^d$, it follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \|   \varphi_{P,\Sigma} -  \varphi_{F_0,\Sigma_0}         \|_{ \mathbb{B}(T_n) } \leq M T_n^{1.5} n^{ \gamma} \epsilon_n, \| \Sigma^{-1} \| \leq M^2  \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2).
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

It remains to examine the bias from truncating the $L^2$ norm to the set $\mathbb{B}(T_n)$. Suppose $ \|  \Sigma^{-1}  \| \leq M^2  \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}  $ holds. It follows that there exists a $c > 0$ for which $  \lambda_1(\Sigma ) \geq c (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} $ holds. From the definition of $\sigma_n^2$, we have  $T_n^2 (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} \asymp  \log n$. It follows that there exists a universal constant $C > 0$ such that

\begin{align*}
\| \big( {\varphi}_{F_0,\Sigma_0}   -   {\varphi}_{P,\Sigma} \big) \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 & \leq  2 \|  {\varphi}_{F_0,\Sigma_0}    \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 +  2 \|     {\varphi}_{P,\Sigma}  \mathbbm{1} \{ \| t \|_{\infty} > T_n  \}   \|_{L^2}^2 \\ & \leq 2 \int_{\| t \|_{\infty} > T_n}  e^{- t' \Sigma_0 t} dt + 2 \int_{\| t \|_{\infty} > T_n} e^{- t' \Sigma t} dt \\ & \leq 2 \int_{\| t \|_{\infty} > T_n}  e^{- t' \Sigma_0 t} dt + 2 \int_{\| t \|_{\infty} > T_n} e^{- c \| t \|^2  \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa}   } dt \\ & \leq D \bigg[  e^{- C T_n^2} T_n^{d-2} +  \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa}   e^{- C T_n^2 \sigma_n^{-2} (n \epsilon_n^2)^{-1/\kappa} } T_n^{d-2}   \bigg ] .
\end{align*}

From substituting $T_n^2 (n \epsilon_n^2)^{-1/\kappa} \sigma_n^{-2} \asymp  \log n$, the preceding bound reduces to $D n^{- 2 K  }$ for some constant $ K \in (0,1/2]$ (that depends on, among other factors, the smallest eigenvalue of $\Sigma_0$).

With constant $K$ as specified above, it follows that for any $C' > 0$, there exists $ M > 0$ such that
$$ \nu_{\alpha,G} \bigg(  \| \varphi_{F_0,\Sigma_0}  -  \varphi_{P,\Sigma}      \|_{L^2} \leq M T_n^{1.5} n^{\gamma} \epsilon_n  + n^{-K}  \:     \big|\mathcal{Z}_n,T_n  \bigg) \geq 1- \exp(-C' n \epsilon_n^2)  $$

holds with $\mathbb{P}$ probability approaching $1$. The claim follows from observing that
$$ T_n^{1.5} n^{\gamma} \epsilon_n \asymp n^{-1/2 + \gamma} (\log n)^{\lambda/2 + 3/4}.  $$

The proof proceeds through several steps which we outline below. We use $D > 0$ as a generic universal constant that may change from line to line. For ease of notation, we suppress the dependence of $G = G_n$ on $n$.

Let $\beta = \max \{ 0 , d-2 \}$ and define
\begin{align*}
\alpha_n^2 =  \big( \epsilon_n^2 T_n^{- \beta} \big)^{\frac{1}{\zeta + s}}  \; \; , \; \; \epsilon_n^2 = \frac{(\log n)^{\lambda + d/2}}{n^{\frac{2(s+ \zeta) - \beta}{2(s+ \zeta) + d - \beta}}} \; \; , \; \;       \lambda = \begin{cases}
\chi^{-1}(d +2)  & \chi < 2 \\ d/ \chi + 1 & \chi \geq 2.
\end{cases}
\end{align*}

First, we derive a lower bound for the normalizing constant of the posterior measure.  Specifically, we aim to show that there exists a $C > 0 $ such that
\begin{align}
\label{t1lb}  \int  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \geq \exp \big(   - C n    \epsilon_{n}^2   \big)
\end{align}

holds with $\mathbb{P}$ probability approaching $1$.

As $\mathcal{H}^{d-1}( \partial \mathbb{B}(T_n)) \leq D T_n^{d-1}$, Lemma <ref> implies that
\begin{align*}
&    \int_{\mathbb{B}(T_n)}   \|  \widehat{\varphi}_{Y_1,Y_2}(t) - \varphi_{Y_1,Y_2}(t)   \|^2 dt \leq D \frac{  T_n^{d} \log(T_n)  }{n}  \; ,  \\ &  \int_{ \partial \mathbb{B}(T_n)}   \|  \widehat{\varphi}_{Y_1,Y_2}(t) - \varphi_{Y_1,Y_2}(t)   \|^2 d \mathcal{H}^{d-1}(t) \leq D \frac{  T_n^{d-1} \log(T_n)  }{n} .
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$. On this set, it follows that
\begin{align*}
& \int  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \\ &  \geq \exp(-n D \epsilon_n^2) \int  \exp \bigg( - \frac{n D}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       -  \varphi_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma).
\end{align*}

Observe that $\varphi_{Y_1,Y_2} = \varphi_{Y_2} \nabla \log \varphi_{X}$. By Condition <ref>, there exists universal constants $ \chi , C, M    < \infty$ and a mixing distribution $S_{\alpha_n}$ supported on the cube $ I_{n} = [-C  (\log \epsilon_n^{-1})^{1/\chi},C  (\log \epsilon_n^{-1})^{1/\chi} ]^d$ that satisfies
$$ \| \varphi_{Y_2} \big( \nabla \log \varphi_{X} - \nabla \log \varphi_{S_{\alpha_n}, \alpha_n^2 I}    \big)  \|_{\partial,\; \mathbb{B}(T_n)}^2 \leq D T_n^{\beta}  \alpha_n^{2( \zeta + s)} \leq D \epsilon_n^2 .  $$

It follows that
\begin{align*}
& \int  \exp \bigg( - \frac{nD}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - {\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \\ &  \geq \exp(-n D \epsilon_n^2) \int  \exp \bigg( - \frac{n D}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       -  \varphi_{Y_2} \nabla \log (\varphi_{S_{\alpha_n}, \alpha_n^2 I} )   \|_{\partial,\;\mathbb{B}(T)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma).
\end{align*}

By Condition <ref> and the definition of $(\alpha_n,T_n)$, there exists a $C_1 >0$ such that
$$ \inf_{\| t \|_{\infty} \leq T_n }  \left| \varphi_{S_{\alpha_n}}(t) \right| \geq n^{-C_1} .$$

Next, we show that $ S_{\alpha_n} $ can be suitably approximated by a discrete measure. Fix any $\gamma > 1$ such that $ \epsilon_n^{2\gamma} T_n^{d} \leq D \epsilon_n^2  $.  By Lemma <ref>, there exists a discrete measure $F' = \sum_{i=1}^N p_i \delta_{\mu_i} $ with at most $ N = D  (\log \epsilon_n^{-1})^{ d/\chi } T_n^d  $ support points on $I_n$ such that
\begin{align*}
&   \sup_{\| t \|_{\infty} \leq T_n } \left|  \varphi_{ S_{\alpha_n}  }(t) - \varphi_{F'}(t)  \right| \leq    n^{-C_1} \epsilon_n^{\gamma}   \\ &  \sup_{\| t \|_{\infty} \leq T_n} \|  \nabla \varphi_{S_{\alpha_n} } (t) - \nabla \varphi_{F'}(t)   \| \leq n^{-C_1} \epsilon_n^{\gamma}.
\end{align*}

From the final claim of Lemma <ref>, we can also assume without loss of generality that the support points satisfy $  \inf_{k \neq j} \| \mu_k - \mu_j   \| \geq \epsilon_n^{L_2}  $ for some $L_2 > 0$. By Condition <ref>, $ \sup_{\| t \|_{\infty} \leq T_n}  \| \nabla \log \varphi_{S_{\alpha_n}}  (t) \|  \leq D $. From the preceding bounds, it follows that

\begin{align*}
&  \sup_{ \| t \|_{\infty} \leq T_n} \| \nabla \log \varphi_{S_{\alpha_n}}  (t) \| \frac{\left|  \varphi_{S_{\alpha_n}} (t) -  \varphi_{F'}(t)  \right|}{\left| \varphi_{S_{\alpha_n}} (t) \right|}     \leq   D \epsilon_n^{\gamma}   \\ &  \sup_{ \| t \|_{\infty}  \leq T_n} \frac{ \| \nabla \varphi_{S_{\alpha_n}}(t) -  \nabla \varphi_{F'}(t)\| }{ \left|  \varphi_{S_{\alpha_n}}(t)  \right|}  \leq D \epsilon_n^{\gamma}.
\end{align*}

Since $\epsilon_n \downarrow 0$, observe that the preceding bound also implies
$$  \left|   \varphi_{F'}(t)  \right| \geq \left| \varphi_{S_{\alpha_n}}(t)    \right| - \left| \varphi_{S_{\alpha_n}}(t) - \varphi_{F'}(t) \right| \geq   \left| \varphi_{S_{\alpha_n}}(t)    \right| - D \epsilon_n \left| \varphi_{S_{\alpha_n}}(t)    \right|  \geq \frac{1}{2} \left| \varphi_{S_{\alpha_n}}(t)    \right|   $$

for all sufficiently large $n$ and $\|t \| \leq T_n$.
It follows that
\begin{align*}
& \sup_{\| t \|_{\infty} \leq T_n}  \| \nabla \log  \varphi_{S_{\alpha_n}}  (t) - \nabla \log \varphi_{F'}(t)     \| \\ & = \sup_{\| t \|_{\infty} \leq T_n}  \bigg \|  \nabla \log \varphi_{S_{\alpha_n}} (t) \frac{\varphi_{F'} (t) - \varphi_{S_{\alpha_n}} (t) }{ \varphi_{F'}(t) }     +  \frac{ \nabla  \varphi_{S_{\alpha_n}} (t)  - \nabla  \varphi_{F'} (t)  }{\varphi_{F'}(t)  }         \bigg \| \\ & \leq \sup_{\| t \|_{\infty} \leq T_n} \bigg(  \left|  \frac{\varphi_{F'}(t) - \varphi_{S_{\alpha_n}}(t) }{ \varphi_{F'} (t) }    \right| \|  \nabla \log  \varphi_{S_{\alpha_n}} (t)  \| \: +  \: \bigg \|  \frac{ \nabla  \varphi_{S_{\alpha_n}} (t)  - \nabla  \varphi_{F'} (t)  }{\varphi_{F'}(t)  }     \bigg \|        \bigg)  \\ & \leq  2 \sup_{\| t \|_{\infty} \leq T_n} \bigg(  \left|  \frac{\varphi_{F'}(t) - \varphi_{S_{\alpha_n}}(t) }{ \varphi_{S_{\alpha_n}} (t) }    \right| \|  \nabla \log  \varphi_{S_{\alpha_n}} (t)  \| \: +  \: \bigg \|  \frac{ \nabla  \varphi_{S_{\alpha_n}} (t)  - \nabla  \varphi_{F'} (t)  }{\varphi_{S_{\alpha_n}}(t)  }     \bigg \|        \bigg) \\ & \leq D \epsilon_n^{\gamma}.
\end{align*}

Fix any $L_3 > L_2$  sufficiently large such that
\begin{align*}  n^{C_1} T_n \epsilon_n^{L_3} \leq \epsilon_n^{\gamma}  \; , \; \; n^{C_1} \sqrt{n}  \epsilon_n \sqrt{\log n} \epsilon_n^{L_3/2}  \leq \epsilon_n^{\gamma} \; , \;   n^{C_1} T_n (\log \epsilon_n^{-1})^{1/\chi} \epsilon_n^{L_3} \leq \epsilon_n^{\gamma}.
\end{align*}

Define $V_i = \{ t \in I_{n} :  \| t - \mu_i  \| \leq  \epsilon_n^{L_3}  \} $ for $i=1,\dots,N$ and set $V_0 = \R^d \setminus \bigcup_{i=1}^N V_i$. From the definition of the $\{\mu_i \}_{i=1}^N$, it follows that $\{V_0, V_1, \dots , V_N  \}$ is a disjoint partition of $\R^d$. By Lemma <ref>, for any distribution $P$ that satisfies $  \int_{\R^d} \| x \|^2 dP(x) \leq n \epsilon_n^2 \log n  $ and $ \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3} $, we have that
\begin{align*}
&  \sup_{\| t \|_{\infty} \leq T_n}  \|  \nabla  \log \varphi_{F'}(t)  \|  \frac{ \left|  \varphi_{P}(t) - \varphi_{F'}(t)  \right|}{\left|  \varphi_{F'}(t)  \right|} \leq  D \epsilon_n^{\gamma} \; , \; \sup_{\| t \|_{\infty} \leq T_n} \frac{ \|  \nabla \varphi_{P} (t) - \nabla \varphi_{F'}(t)   \|}{\left| \varphi_{F'}(t)  \right|} \leq D \epsilon_n^{\gamma}.
\end{align*}

For all such $P$, an analogous argument to the bound for $S_{\alpha_n}$ and $F'$ implies that
\begin{align*}
\sup_{\| t \|_{\infty} \leq T_n}  \| \nabla \log  \varphi_{ P }  (t) - \nabla \log \varphi_{F'}(t)     \| \leq D \epsilon_n^{\gamma}.
\end{align*}

From combining all the preceding bounds, observe that  all such $P$  also satisfy
\begin{align*} &
\sup_{\| t \|_{\infty} \leq T_n}  \| \nabla \log  \varphi_{ P }  (t) - \nabla \log \varphi_{S_{\alpha_n}}(t)     \| \leq D \epsilon_n^{\gamma}
\\ &
\sup_{\| t \|_{\infty} \leq T_n } \|  \nabla \log \varphi_P(t)   \|   \leq D \epsilon_n^{\gamma} + \sup_{\| t \|_{\infty} \leq T_n } \|  \nabla \log \varphi_{S_{\alpha_n}}(t)  \| \leq D.  \end{align*}

Given any positive definite $\Sigma$ with $\| \Sigma \| \leq D \alpha_n^2 $ the preceding bound also implies that
\begin{align*}
\sup_{\| t \|_{\infty} \leq T_n } \|  \nabla \log \varphi_{P,\Sigma}(t)   & \| \leq \sup_{\| t \|_{\infty} \leq T_n } \bigg[ \|  \nabla \log \varphi_{P}(t)   \| + \|  \nabla \log \varphi_{\Sigma}(t)   \|   \bigg] \\ & \leq D  \big( 1 + T_n \alpha_n^2 \big) \\ & \leq D \;,
\end{align*}

where the last inequality follows from $T_n \alpha_n^2 \lessapprox 1$. By Lemma <ref>, we have that
\begin{align*}
&    \int_{\mathbb{B}(T_n)}   \|  \widehat{\varphi}_{Y_2}(t) - \varphi_{Y_2}(t)   \|^2 dt \leq D \frac{  T_n^{d} \log(T_n)  }{n}  \; ,  \\ &  \int_{ \partial \mathbb{B}(T_n)}   \|  \widehat{\varphi}_{Y_2}(t) - \varphi_{Y_2}(t)   \|^2 d \mathcal{H}^{d-1}(t) \leq D \frac{  T_n^{d-1} \log(T_n)  }{n}
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$. On this set, when $(P,\Sigma)$ satisfy the preceding requirements, we have that

\begin{align*}
\|  \widehat{\varphi}_{Y_2} \nabla \log \varphi_{P,\Sigma} - {\varphi}_{Y_1,Y_2}   \|_{\mathbb{B}(T_n)}^2 & \leq D \bigg[  \|  {\varphi}_{Y_2} \nabla \log \varphi_{P,\Sigma} -{\varphi}_{Y_1,Y_2}   \|_{\mathbb{B}(T_n)}^2  +  \frac{T_n^{d} \log(T_n) }{n}  \bigg] \\ & \leq D \bigg[  \|  {\varphi}_{Y_2} \nabla \log \varphi_{P,\Sigma} -{\varphi}_{Y_1,Y_2}   \|_{\mathbb{B}(T_n)}^2  +  \epsilon_n^2 \bigg].
\end{align*}

Define the sets

\begin{align*}
\Omega_n =  \bigg \{ \Sigma \in \mathbf{S}_{+}^d  : \lambda_j(\Sigma) \in \bigg[  \frac{ \alpha_n^2 }{1 + \epsilon_n^{\gamma} T_n^{-1}    }  ,      \alpha_n^2   \bigg ]  \; \; \forall \; j=1,\dots,d.   \bigg \}
\end{align*}

$$ \mathcal{G}_n = \bigg \{ (P,\Sigma) : \Sigma \in \Omega_n , \int_{\R^d} \| x \|^2 dP(x) \leq n \epsilon_n^2 \log(n) \: ,   \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3}   \bigg  \}.  $$

From combining the all the preceding bounds, it follows that
\begin{align*}
&  \int  \exp \bigg( - nD \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       -  \varphi_{Y_2} \nabla \log (\varphi_{S_{\alpha_n}, \alpha_n^2 I} )   \|_{\partial,\;\mathbb{B}(T)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \\ & \geq  \int_{\mathcal{G}_n}  \exp \bigg( - nD \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       -  \varphi_{Y_2} \nabla \log (\varphi_{S_{\alpha_n}, \alpha_n^2 I} )   \|_{\partial,\;\mathbb{B}(T)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \exp \big(  -nD \epsilon_n^2 \big) \int_{ \mathcal{G}_n }   \exp \bigg( - nD \| {\varphi}_{Y_2} \big[ \nabla \log( \varphi_{P,\Sigma} )  - \nabla \log (\varphi_{S_{\alpha_n}, \alpha_n^2 I} ) \big]   \|_{\partial,\;\mathbb{B}(T)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma).
\end{align*}

For every $(P,\Sigma) \in \mathcal{G}_n$, the preceding bounds imply that
\begin{align*}
&  \sup_{\|t \|_{\infty} \leq T_n } \|  \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{S_{\alpha_n}, \alpha_n^2 I}    \| \\ &  = \sup_{\|t \|_{\infty} \leq T_n } \|   \big (\nabla \log \varphi_{P} - \nabla \log \varphi_{S_{\alpha_n}} \big) + \big(  \nabla \log \varphi_{\Sigma} -\nabla \log \varphi_{ \alpha_n^2 I}    \big)  \| \\ & \leq \sup_{\|t \|_{\infty}\leq T_n } \|   \nabla \log \varphi_{P} - \nabla \log \varphi_{S_{\alpha_n}}  \| + \sup_{\|t \|_{\infty}\leq T_n } \|   \nabla \log \varphi_{\Sigma} - \nabla \log \varphi_{\alpha_n^2 I}  \| \\ & \leq D \epsilon_n^{\gamma} + D \| \Sigma - \alpha_n^2 I \| T_n \\ & \leq D \epsilon_n^{\gamma}.
\end{align*}

Since $ \| \varphi_{Y_2}  \|_{L^2} < \infty$ and $\gamma > 1$ is such that $ T_n^d \epsilon_n^{2 \gamma} \leq D \epsilon_n^2$, it follows (from the change of variables and coarea argument in cov) that

\begin{align*}
& \int_{ \mathcal{G}_n }  \exp \bigg( - nD    \|  {\varphi}_{Y_2} \big( \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{S_{\alpha_n}, \alpha_n^2 I}      \big)  \|_{\partial,\;\mathbb{B}(T_n)}^2        \bigg)  d \nu_{\alpha,G} (P,\Sigma) \\ & \geq \exp \big(  -nD \epsilon_n^2 \big) \int_{ \mathcal{G}_n }  d \nu_{\alpha,G} (P,\Sigma).
\end{align*}

Define the sets
\begin{align*}
&  \mathcal{G}_{n,1} = \bigg \{ (P,\Sigma) : \Sigma \in \Omega_n  \: ,   \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3}   \bigg  \} \; , \\ &  \mathcal{G}_{n,2} = \bigg \{ (P,\Sigma) : \Sigma \in \Omega_n  , \int_{\R^d} \| x \|^2 dP(x) > n \epsilon_n^2 \log(n) \: ,   \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3}   \bigg  \} .
\end{align*}

Observe that $\mathcal{G}_n = \mathcal{G}_{n,1} \setminus \mathcal{G}_{n,2} $. It follows that
\begin{align*}
\int_{ \mathcal{G}_n }  d \nu_{\alpha,G} (P,\Sigma) = \int_{ \mathcal{G}_{n,1} }  d \nu_{\alpha,G} (P,\Sigma)  - \int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G} (P,\Sigma) .
\end{align*}

For the second term, observe that
\begin{align*}
\int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G} (P,\Sigma) &  \leq \int_{ P : \int_{\R^d} \| x \|^2 dP(x) > n \epsilon_n^2 \log(n)  }  d \nu_{\alpha,G} (P,\Sigma) \\ & \leq  \int_{ P : \int_{\R^d} \| x \|^2 dP(x) > n \epsilon_n^2 \log(n)  }  d \text{DP}_{\alpha}(P) \\ &  \leq \exp(-D n \epsilon_n^2 \log n) \; ,
\end{align*}

where the second inequality is due to $\nu_{\alpha,G}$ being a product measure $\nu_{\alpha,G} = \text{DP}_{\alpha} \otimes G$ and the third inequality follows from an application of Lemma <ref>.

For the first term, we have that
\begin{align*}
\int_{\mathcal{G}_{n,1}} d \nu_{\alpha,G} (P,\Sigma) = \int_{ \Sigma : \Sigma \in \Omega_n } \int_{P : \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3}    } d  \text{DP}_{\alpha}(P)  d G(\Sigma).
\end{align*}

As $ \text{DP}_{\alpha} $ is constructed using a Gaussian base measure $\alpha $, it is straightforward to verify that $
\inf_{j=1}^N \alpha(V_j) \geq C \epsilon_n^{L_3 d} \exp(- C' ( \log \epsilon_n^{-1})^{2/\chi}  ) $ for universal constants $C,C' > 0$. By definition of $ \text{DP}_{\alpha}$, $(P(V_1),\dots, P(V_N)) \sim \text{Dir}(N, \alpha(V_1), \dots,  \alpha(V_N))$. As $N = D \{ \log ( \epsilon_n^{-1} )  \}^{d/\chi} T_n^d$, an application of <cit.> implies
\begin{align*}
\int_{P : \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3}    } d  \text{DP}_{\alpha}(P) \geq C \exp \big (  - C' T_n^d  ( \log \epsilon_n^{-1} )^{d/\chi + \max \{2/ \chi,1   \}  }   \big)  & = C \exp \big (  - C' T_n^d ( \log \epsilon_n^{-1} )^{\lambda  }   \big) \\ & \geq  C \exp \big (  - C''  n \epsilon_n^2   \big) .
\end{align*}

It remains to bound the outer integral. The law of $G= G_n$ is given by $\Omega / \sigma_n^2$ where $\Omega \sim L$ and $L$ is a probability measure on  $\mathbf{S}_+^d$ that  satisfies Assumption <ref>. By Assumption <ref> and the definition of $(\alpha_n^2,\sigma_n^2, \epsilon_n^2)$, there exists a universal constant $C,C',C'' > 0$ such that
\begin{align*}
\int_{  \Sigma \in \Omega_n } d G(\Sigma) = \int_{  \Sigma \in \sigma_n^2 \Omega_n } d L(\Sigma) \geq C \exp \big( - C' \sigma_n^{-2 \kappa} \alpha_n^{-2 \kappa}   \big) \geq C \exp ( -C'' n \epsilon_n^2  ).
\end{align*}

It follows that
\begin{align*}
\int_{ \mathcal{G}_n }  d \nu_{\alpha,G} (P,\Sigma) & = \int_{ \mathcal{G}_{n,1} }  d \nu_{\alpha,G} (P,\Sigma)  - \int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G} (P,\Sigma)  \\ & \geq C \exp(- C'' n \epsilon_n^2) - \exp(-D n \epsilon_n^2 \log n) \\ & \geq \exp(-Dn \epsilon_n^2).
\end{align*}

The  estimate for the lower bound of the normalizing constant follows from combining all the preceding bounds.

Next, we establish a preliminary local concentration bound under the prior. Observe that for any $E > 0$, we have

\begin{align*}
&  \int \limits_{ (P,\Sigma) : \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2     > 2E \epsilon_n^2 }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2           \bigg) d \nu_{\alpha,G} (P,\Sigma) \\ &  \leq \exp \big( - n E \epsilon_n^2    \big).
\end{align*}

The law of $G$ is given by $\Sigma / \sigma_n^2$ where $\Sigma \sim L$ and $L$ is a probability measure on  $\mathbb{S}_+^d$ that  satisfies Assumption <ref>. By Assumption <ref>, it follows that for every $E' > 0$, there exists $E > 0$ such that

\begin{align*}
\int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\Sigma) = \int_{\Sigma : \| \Sigma^{-1}  \| > E (n \epsilon_n^2)^{1/ \kappa}  }   d L(\Sigma) \leq \exp \big(  - E' n \epsilon_n^2  \big).
\end{align*}

As the prior is a product measure $\nu_{\alpha,G} = \text{DP}_{\alpha} \otimes G$ and  $ \| \widehat{\varphi}_{Y}    - \widehat{\varphi}_{\epsilon} {\varphi}_{P,\Sigma}      \|_{\mathbb{B}(T_n)}^2  \geq 0$, the preceding bound implies
\begin{align*}
&  \int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2            \bigg) d \nu_{\alpha,G} (P,\Sigma)  \\ & \leq  \int_{\Sigma : \| \Sigma^{-1}  \| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\Sigma)  \\ & \leq \exp \big(  - E' n \epsilon_n^2  \big).
\end{align*}

From combining the preceding bounds, it follows that for any $E' > 0$ we can pick $E > 0$ sufficiently large such that
\begin{align*} \int \limits_{\substack{(P,\Sigma) : \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2    \leq E \epsilon_n^2 , \\ \| \Sigma^{-1} \| \leq E \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}}}
\exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )      & - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2            \bigg)    d \nu_{\alpha,G} (P,\Sigma) \\ & \geq 1-\exp \big( - E' n \epsilon_n^2   \big) .
\end{align*}

We prove the main statement of the theorem. From the bounds derived in steps $(i)$ and $(ii)$, it follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \| \widehat{\varphi}_{Y_2} \nabla \log( \varphi_{P,\Sigma} )       - \widehat{\varphi}_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}^2    \leq M^2 \epsilon_n^2, \| \Sigma^{-1} \| \leq M^2 \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2)
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

By an application of Lemma <ref>, we have that
\begin{align*}
\|  \widehat{\varphi}_{Y_1,Y_2}  - \varphi_{Y_1,Y_2}   \|_{\partial,\;\mathbb{B}(T)}    \leq D \epsilon_n
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

Since $\varphi_{Y_1,Y_2} = \varphi_{Y_2} \nabla \log  \varphi_{X} $ and  $  \sup_{\| t \|_{\infty} \leq T_n} \| \nabla \log  \varphi_{X} \| \leq D  $, an application of Lemma <ref> implies that
\begin{align*}
\|   \widehat{\varphi}_{Y_2}  \nabla \log \varphi_{X}  - \varphi_{Y_1,Y_2}  \|_{\partial,\;\mathbb{B}(T)}    \leq D  \epsilon_n
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

From combining the preceding bounds, it follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \nu_{\alpha,G} \bigg(  \| \widehat{\varphi}_{Y_2} \big( \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{X} \big)         \|_{\partial,\;\mathbb{B}(T)}^2     \leq M^2 \epsilon_n^2, \| \Sigma^{-1} \| \leq M^2 \sigma_n^2  (n \epsilon_n^2)^{1/\kappa} \:    \bigg  |\mathcal{Z}_n,T_n \bigg )
\\ & \geq 1- \exp(-C' n \epsilon_n^2)
\end{align*}

holds with $\mathbb{P}$ probability approaching $1$.

An application of Lemma <ref> implies that
\begin{align*}
\sup_{\| t \|_{\infty} \leq T_n}   \frac{  \left| \widehat{\varphi}_{Y_2}(t)  - \varphi_{Y_2}(t)    \right| }{\left| \varphi_{Y_2}(t) \right|} \leq D T_n^{\zeta} \frac{\sqrt{\log n}}{\sqrt{n}}
\end{align*}

with $\mathbb{P}$ probability approaching $1$. As the quantity on the right converges to zero, we also have

\begin{align*}
\left|   \widehat{\varphi}_{Y_2}(t)    \right|  & \geq  \left|   {\varphi}_{Y_2}(t)    \right| -  \left| \widehat{\varphi}_{Y_2}(t)  - \varphi_{Y_2}(t)    \right|  \geq \frac{1}{2} \left|   {\varphi}_{Y_2}(t)   \right|
\end{align*}

uniformly over the set $ \{ \| t \|_{\infty} \leq T_n \}$. It follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}