& \nu_{\alpha,G} \bigg(  \|  \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{X}         \|_{\partial,\;\mathbb{B}(T)}^2     \leq M^2 T_n^{2 \zeta} \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 $\varphi_{X}$ are characteristic functions, they satisfy the initial value condition $  \log \varphi_{P,\Sigma} (0) = \log \varphi_{X}(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_{X}(z) = \int_{0}^1 \langle \nabla \log \varphi_{P,\Sigma} (tz) - \nabla \log \varphi_{X}(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_{X}(z)  \right|^2 \leq  \| z \|^2 \int_{0}^1 \| \nabla \log \varphi_{P,\Sigma}(tz) - \nabla \log \varphi_{X} (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_{X}(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_{X} (tz) \|^2 dt d \mathcal{H}^{d-1}(z) .
\end{align*}

Suppose $(P,\Sigma)$ is such that  $\|  \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{X}         \|_{\partial,\;\mathbb{B}(T)}^2     \leq M^2 T_n^{2 \zeta} \epsilon_n^2$. By definition of the metric, this means that
\begin{align*}
& \int_{\mathbb{B}(T_n) } \| \nabla \log \varphi_{P,\Sigma} - \nabla \log \varphi_{X}    \|^2 dt  \leq M^2 T_n^{2 \zeta} \epsilon_n^2 \; , \\ & \int_{\partial \mathbb{B}(T_n)} \int_{0}^1   \| \nabla \log \varphi_{P,\Sigma}(tz) - \nabla \log \varphi_{X} (tz) \|^2 dt d \mathcal{H}^{d-1}(z) \leq M^2 T_n^{2 \zeta} \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_{X}  \|_{\mathbb{B}(T_n)}^2  \leq D M^2 T_n^{3 + 2 \zeta} \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_{X}         \|_{ \mathbb{B}(T_n) } \leq M T_n^{1.5 + \zeta}  \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 \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, it follows that for every $C' > 0$, there exists a $ M > 0$ such that
$$ \nu_{\alpha,G} \bigg(  \| \varphi_{X}  -  \varphi_{P,\Sigma}      \|_{L^2} \leq M \big( T_n^{1.5 + \zeta} \epsilon_n + T_n^{-p}   \big)   \:     \bigg|\mathcal{Z}_n,T_n  \bigg) \geq 1- \exp(-C' n \epsilon_n^2) . $$

holds with $\mathbb{P}$ probability approaching $1$. The claim follows from observing that
\begin{align*}
T_n^{3 + 2\zeta} \epsilon_n^2 \asymp  \frac{(\log n)^{\lambda + d/2 + \zeta + 3/2 }}{ n^{\frac{2s - \beta - 3}{2(s+ \zeta) + d - \beta}} }.
\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}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T) }^2        \bigg)  d \nu_{\alpha,G} (P,\sigma^2) \geq \exp \big(   - C n    \epsilon_{n}^2   \big)
\end{align}

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

Note that $\left|t' \mathbf{A}_k \right| \leq \sqrt{d} \| t \|_{\infty} \| \mathbf{A}_k \| \leq \sqrt{d}T_n \| \mathbf{A}_k \|   $ uniformly over $t \in \mathbb{B}(T)$. Since the preceding constant is finite, it suffices to work under the setting where $\left| t' \mathbf{A}_k \right| \leq D T_n$. Fix  $\epsilon > 0$ sufficiently small. By Condition <ref>, the mixing distribution $F_0$ satisfies $F_0( t \in \R :  \left| t \right|  > z) \leq C \exp(- C' z^{\chi})$. Hence, there exists  a universal constant $R > 0$ such that the cube $I =[-R  (\log \epsilon^{-1})^{1/\chi},R  (\log \epsilon^{-1})^{1/\chi} ] $ satisfies $ 1- F_0(I) \leq D \epsilon$. Denote the probability measure induced from the restriction of $F_0$ to $I$ by
$$ \overline{F}_0(A) = \frac{F_0(A \cap I)}{F_0(I)}  \; \; \; \; \; \forall \; \; \text{Borel}  \; A \subseteq \R. $$

Observe that
\begin{align*}
\sup_{t \in \R} \left| \varphi_{F_0} (t) -  \varphi_{\overline{F}_0}(t) \right| =  \sup_{t \in \R} \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 $  r=1,2 $  and all sufficiently large $ M > 0$, the tail bound on $F_0$ implies

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

For $r=1,2$ it follows that

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

We can write $(\log \varphi_{F_0})''  -  (\log \varphi_{ \overline{F}_0})'' $ as
\begin{align*}
&  (\log \varphi_{F_0})'' -  (\log \varphi_{ \overline{F}_0})''  \\ & = (\log \varphi_{F_0})'' \frac{( \varphi_{\overline{F}_0}^2 - \varphi_{F_0}^2    )}{ \varphi_{\overline{F}_0}^2  } +  \frac{  \varphi_{F_0} (  \varphi_{F_0}''  - \varphi_{\overline{F}_0}''    )      }{ \varphi_{\overline{F}_0}^2 }  +   \frac{  \varphi_{\overline{F}_0}'' (  \varphi_{F_0} - \varphi_{\overline{F}_0}    )        }{ \varphi_{\overline{F}_0}^2 }        + \frac{ (\varphi_{\overline{F}_0}')^2  - (\varphi_{{F}_0}')^2   }{\varphi_{\overline{F}_0}^2}.
\end{align*}

Since $ \inf_{ \left|t \right| \leq D T_n }  \left| \varphi_{F_0}(t) \right| \geq C \exp(-C' T_n^2 ) $ for some $C,C' > 0$ and $T_n^2 \lessapprox \log(n) $, there exists a $C_1 > 0$ such that
$$ \inf_{ \left| t \right| \leq DT_n }  \left| \varphi_{F_0}(t) \right| \geq n^{-C_1} .$$

Let $L > 1$ be such that $ n^{2 C_1} (\log \epsilon_{n}^{-L})^{2/\chi} \epsilon_n^L \leq \epsilon_n$. 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} ] $  for some universal constant $E > 0$. Since $\epsilon_n \downarrow 0$, we also have that
$$  \left|   \varphi_{\overline{F}_0}(t)  \right| \geq \left| \varphi_{F_0}(t)    \right| - \left| \varphi_{F_0}(t) - \varphi_{\overline{F}_0}(t) \right| \geq   \left| \varphi_{F_0}(t)    \right| - D \epsilon_n \left| \varphi_{F_0}(t)    \right|  \geq \frac{1}{2} \left| \varphi_{F_0}(t)    \right|  \geq \frac{1}{2} n^{-C_1}  $$

for all sufficiently large $n$ and $ \left| t \right| \leq DT_n$. Since $\left| (\log \varphi_{F_0})''  \right| , \left|  \varphi_{F_0}'  \right| , \left|  \varphi_{F_0}''  \right|  $ are bounded by a universal constant $D$ and  $ n^{2 C_1} (\log \epsilon_{n}^{-L})^{2/\chi} \epsilon_n^L \leq \epsilon_n$, the preceding expression for $(\log \varphi_{F_0})''  -  (\log \varphi_{ \overline{F}_0})''$ implies that
\begin{align*}
\sup_{\left| t \right| \leq D T_n}  \left| (\log \varphi_{F_0})'' (t)  -  (\log \varphi_{ \overline{F}_0})'' (t) \right| \leq D  n^{2 C_1} (\log \epsilon_{n}^{-L})^{2/\chi} \epsilon_n^L \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_{ \left| t \right| \leq DT_n } \left| \partial_{t}^r  \varphi_{\overline{F}_0}(t) - \partial_{t}^r \varphi_{F'}(t)  \right| \leq    n^{-2C_1} \epsilon_n   \; \; \; \; \; \; \; r=0,1,2.
\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 $L_2 > 0$. Observe that $2 \inf_{ \left| t \right| \leq D T_n }  \left| \varphi_{\overline{F}_0}(t)  \right|  \geq  \inf_{\left| t \right| \leq DT_n }  \left| \varphi_{F_0}(t) \right| \geq n^{-C_1}$ and

\begin{align*}
\sup_{\left| t \right| \leq DT_n }  \left|  (\log \varphi_{ \overline{F}_0})''  (t)  \right|  & \leq \sup_{\left| t \right| \leq D T_n}  \left| (\log  \varphi_{F_0} )''  (t) -  (\log \varphi_{\overline{F}_0} )''(t)     \right| +  \sup_{\left| t \right| \leq DT_n }  \left| ( \log \varphi_{ F_0 }  )''(t)  \right| \\ & \leq D \big( \epsilon_n + D \big) \\ & \leq D .
\end{align*}

From the preceding bounds and an analogous expression to that of $(\log \varphi_{F_0})''  -  (\log \varphi_{ \overline{F}_0})'' $, it follows that
\begin{align*}
\sup_{\left| t \right| \leq  DT_n}  \left| ( \log  \varphi_{ \overline{F}_0 } )''  (t) - ( \log \varphi_{F'})''(t)     \right| \leq D \epsilon_n.
\end{align*}

Fix any $L_3 > L_2$  sufficiently large such that
$$   n^{2 C_1 + 1} \epsilon_n^2 \log(n) \epsilon_n^{L_3/2} \leq \epsilon_n \:,\: n^{2 C_1} (\log \epsilon_n)^{2/ \chi} T_n \epsilon_n^{L_3 }  \leq \epsilon_n .  $$

Define $V_i = \{ t \in I_{n} :  \left| t - \mu_i  \right| \leq  \epsilon_n^{L_3}  \} $ for $i=1,\dots,N$ and set $V_0 = \R \setminus \bigcup_{i=1}^N V_i$. 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 $\R$. By Lemma <ref>, for any distribution $P$ that satisfies $  (\int_{\R} \left| x \right|^4 dP(x))^{1/4} \leq \sqrt{n} \epsilon_n \sqrt{\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_{\left| t \right| \leq T_n}  \left| \partial_{t}^r \varphi_{F'}(t) - \partial_{t}^r \varphi_{P}(t)    \right| \leq n^{-2 C_1} \epsilon_n \; \; \; \; \; \; \;  r=0,1,2.
\end{align*}

For all such $P$, the preceding bound and an analogous expression to that of $(\log \varphi_{F_0})''  -  (\log \varphi_{ \overline{F}_0})''$, yields
\begin{align*}
\sup_{ \left| t \right| \leq  D T_n}  \left| ( \log  \varphi_{ F' } )''  (t) - ( \log \varphi_{P})''(t)     \right| \leq D \epsilon_n.
\end{align*}

From combining all the preceding bounds, observe that all such $P$  also satisfy
\begin{align*} &
\sup_{ \left| t \right| \leq DT_n}  \left| ( \log  \varphi_{ P } )'' (t) -  (\log \varphi_{F_0} )''(t)     \right| \leq D \epsilon_n
\\ &
\sup_{\left| t \right|  \leq D T_n } \left|  ( \log \varphi_P )'' (t)   \right|   \leq D \epsilon_n + \sup_{\left| t \right|  \leq DT_n } \left| ( \log \varphi_{F_0})''(t)  \right| \leq D .
\end{align*}

Given any $\sigma^2 \geq 0$, the preceding bound also implies that
\begin{align*}
\sup_{\left| t \right| \leq D T_n } \left|  (\log \varphi_{P,\sigma^2})''(t) \right|  &  \leq \sup_{\left| t \right| \leq D T_n } \bigg[ \left| ( \log \varphi_{P})''(t)   \right| + \left|  (\log \varphi_{\sigma^2} )''(t)   \right|   \bigg]  \\ &  \leq D  \big( 1 +  \sigma^2 \big).
\end{align*}

Observe that the $(l,k)$ element of $  \widehat{\varphi}_{\mathbf{Y}}^2 (t)    \big[ \nabla \nabla' \log \widehat{ \varphi}_{\mathbf{Y}} (t)          \big]$ is given by
$$   -   \widehat{\varphi}_{\mathbf{Y}}(t)     \E_n[Y_l Y_k e^{\mathbf{i} t' \mathbf{Y}}  ] + \E_n[Y_l e^{\mathbf{i} t' \mathbf{Y}}  ] \E_n[ Y_k e^{\mathbf{i} t' \mathbf{Y}}  ]  .   $$

From this representation and an application of Lemma <ref> and <ref>, we have that
\begin{align*}
\|  \widehat{\varphi}_{\mathbf{Y}}^2 \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}}-  {\varphi}_{\mathbf{Y}}^2 \mathbf{Q}_k^* {\mathcal{V}}_{\mathbf{Y}}     \|_{\mathbb{B}(T)}^2 \leq D \frac{T_n^d \log(T_n)}{n} \leq D \epsilon_n^2.
\end{align*}

with $\mathbb{P}$ probability approaching $1$. On this set, when $P$ is specified as above and $ \sigma^2 \leq D$, we have that

\begin{align*}
& \|  \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k) \big)    \|_{\mathbb{B}(T_n)}^2  \\ & \leq D \bigg[  \|  {\varphi}_{\mathbf{Y}}^2 \big( \mathbf{Q}_k^* {\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k) \big)     \|_{\mathbb{B}(T_n)}^2  +  \epsilon_n^2 \bigg].
\end{align*}

Define the set
$$ \mathcal{G}_n = \bigg \{ (P, \sigma^2) : \left|  \sigma^2 - \sigma_0^2 \right| \leq \epsilon_n^2 , \int_{\R} \left|x \right|^4 dP(x) \leq n^2 \epsilon_n^4 (\log n)^2 \: ,   \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3}   \bigg  \}.  $$

From combining the all the preceding bounds, it follows that
\begin{align*}
&   \int  \exp \bigg( - \frac{n}{2} \|  \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k) \big)    \|_{\mathbb{B}(T_n)}^2       \bigg)  d \nu_{\alpha,G} (P,\sigma^2) \\ & \geq  \int_{ \mathcal{G}_n }  \exp \bigg( - \frac{n}{2} \|  \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k) \big)    \|_{\mathbb{B}(T_n)}^2       \bigg)  d \nu_{\alpha,G} (P,\sigma^2) \\ & \geq \exp \big(  -nD \epsilon_n^2 \big) \int_{ \mathcal{G}_n }  \exp \bigg( - nD    \|  {\varphi}_{\mathbf{Y}}^2 \big( \mathbf{Q}_k^* {\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k) \big)     \|_{\mathbb{B}(T_n)}^2     \bigg)  d \nu_{\alpha,G} (P,\sigma^2).
\end{align*}

Since $ \mathbf{Q}_k^* \mathcal{V}_{\mathbf{Y}}(t) = (\log \varphi_{X})'' (t'\mathbf{A}_k)   $ and $\varphi_{X} = \varphi_{F_0, \sigma_0^2}$, the preceding integral can be expressed as
\begin{align*}
& \int_{ \mathcal{G}_n }  \exp \bigg( - nD    \|  {\varphi}_{\mathbf{Y}}^2 \big( \mathbf{Q}_k^* {\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k) \big)     \|_{\mathbb{B}(T_n)}^2     \bigg)  d \nu_{\alpha,G} (P,\sigma^2) \\ & = \int_{ \mathcal{G}_n }  \exp \bigg( - nD    \|  {\varphi}_{\mathbf{Y}}^2 \big[ (\log \varphi_{F_0,\sigma_0^2})'' (t'\mathbf{A}_k) - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k) \big]     \|_{\mathbb{B}(T_n)}^2     \bigg)  d \nu_{\alpha,G} (P,\sigma^2).
\end{align*}

For every $(P,\Sigma) \in \mathcal{G}_n$, the preceding bounds imply that
\begin{align*}
\sup_{\| t\|_{\infty} \leq  T_n} \left|  (\log \varphi_{F_0,\sigma_0^2})'' (t'\mathbf{A}_k) - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)      \right|     \leq D \epsilon_n.
\end{align*}

Since $ \|  \varphi_{\mathbf{Y}}^2  \|_{L^2} < \infty $, the preceding bound implies that
\begin{align*}
&  \int_{ \mathcal{G}_n }  \exp \bigg( - nD    \|  {\varphi}_{\mathbf{Y}}^2 \big[ (\log \varphi_{F_0,\sigma_0^2})'' (t'\mathbf{A}_k) - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k) \big]     \|_{\mathbb{B}(T_n)}^2     \bigg)  d \nu_{\alpha,G} (P,\sigma^2) \\ & \geq \exp(-nD \epsilon_n^2) \int_{ \mathcal{G}_n } d \nu_{\alpha,G} (P,\sigma^2).
\end{align*}

Define the sets
\begin{align*}
&  \mathcal{G}_{n,1} = \bigg \{ (P, \sigma^2) : \left|  \sigma^2 - \sigma_0^2 \right| \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^2) : \left|  \sigma^2 - \sigma_0^2 \right| \leq \epsilon_n^2 , \int_{\R} \left|x \right|^4 dP(x) >  n^2 \epsilon_n^4 (\log n)^2 \: ,   \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} $. Hence
\begin{align*}
\int_{ \mathcal{G}_n }  d \nu_{\alpha,G} (P,\sigma^2) = \int_{ \mathcal{G}_{n,1} }  d \nu_{\alpha,G} (P,\sigma^2)  - \int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G} (P,\sigma^2) .
\end{align*}

For the second term, observe that
\begin{align*}
\int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G} (P,\sigma^2) &  \leq \int_{ P : \int_{\R} \left| x \right|^4 dP(x) >  n^2 \epsilon_n^4 (\log n)^2 }  d \nu_{\alpha,G} (P,\sigma^2) \\ & \leq  \int_{ P : \int_{\R} \left| x \right|^4 dP(x) > n^2 \epsilon_n^4 (\log n)^2  }  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^2) = \int_{ \sigma^2 : \left| \sigma^2 - \sigma_0^2  \right| \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^2).
\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^{L_3}    } 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. The law of $G$ is given by $\Omega / \sigma_n^2$ where $\Omega \sim L$ and $L$ is a probability measure on  $\R_{+}$ 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^2 : \left| \sigma^2 - \sigma_0^2  \right| \leq  \epsilon_n^2} d G(\sigma^2) = \int_{\sigma^2 : \left| \sigma^2 - \sigma_n^2 \sigma_0^2   \right| \leq \sigma_n^2 \epsilon_n^2  } d L(\sigma^2) \geq C  \exp \big( - C' \sigma_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^2) & = \int_{ \mathcal{G}_{n,1} }  d \nu_{\alpha,G} (P,\sigma^2)  - \int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G} (P,\sigma^2)  \\ & \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^2) : \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T_n) }^2 > 2E \epsilon_n^2 }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \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^2 : \left| \sigma^{-2}  \right| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\sigma^2) = \int_{\sigma^2 : \left| \sigma^{-2}  \right| > E (n \epsilon_n^2)^{1/ \kappa}  }   d L(\sigma^2) \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}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T_n) }^2  \geq 0$, the preceding bound implies
\begin{align*}
&  \int_{\sigma^2 : \left| \sigma^{-2}  \right| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T_n) }^2        \bigg) d \nu_{\alpha,G} (P,\sigma^2)  \\ & \leq  \int_{\sigma^2 : \left| \sigma^{-2}  \right| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\sigma^2)  \\ & \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^2) : \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T_n) }^2 \leq E \epsilon_n^2 , \\ \left| \sigma^{-2} \right| \leq E \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}}}
\exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T_n) }^2        \bigg)    d \nu_{\alpha,G} (P,\sigma^2) \\ & \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}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \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 $1$.

The $(l,k)$ element of $  \widehat{\varphi}_{\mathbf{Y}}^2 (t)  \widehat{\mathcal{V}}_{\mathbf{Y}} (t) $ is given by
$$   -   \widehat{\varphi}_{\mathbf{Y}}(t)     \E_n[Y_l Y_k e^{\mathbf{i} t' \mathbf{Y}}  ] + \E_n[Y_l e^{\mathbf{i} t' \mathbf{Y}}  ] \E_n[ Y_k e^{\mathbf{i} t' \mathbf{Y}}  ]  .   $$

From this representation and an application of Lemma <ref> and <ref>, we have that
\begin{align*}
\|  \widehat{\varphi}_{\mathbf{Y}}^2 \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}}-  {\varphi}_{\mathbf{Y}}^2 \mathbf{Q}_k^* {\mathcal{V}}_{\mathbf{Y}}     \|_{\mathbb{B}(T)} \leq D \epsilon_n.
\end{align*}

Since $ \mathbf{Q}_k^* \mathcal{V}_{\mathbf{Y}}(t) = (\log \varphi_{F_0, \sigma_0^2})'' (t'\mathbf{A}_k)   $  and  $   \left|  \big( \log  \varphi_{F_0,\sigma_0^2} \big)''(t' \mathbf{A}_k)  \right| \leq D  $, an application of Lemma <ref> implies that
\begin{align*}
\|   \widehat{\varphi}_{\mathbf{Y}}^2  \big( \log  \varphi_{F_0,\Sigma_0} \big)''(t' \mathbf{A}_k)  - {\varphi}_{\mathbf{Y}}^2 \mathbf{Q}_k^* {\mathcal{V}}_{\mathbf{Y}}  \|_{\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}_{\mathbf{Y}}^2 \big[ \big(\log  \varphi_{F_0,\sigma_0^2} \big)''(t' \mathbf{A}_k) - \big(\log  \varphi_{P,\sigma^2} \big)''(t' \mathbf{A}_k) \big]         \|_{ \mathbb{B}(T_n) }^2 \leq M^2  \epsilon_n^2, \left| \sigma^{-2} \right| \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 $2 c_0 R  = \gamma < 1/2$, an application of Lemma <ref> implies that
\begin{align*}
\sup_{\| t \|_{\infty} \leq T_n}   \frac{  \left| \widehat{\varphi}_{\mathbf{Y}}^2(t)  - \varphi_{\mathbf{Y}}^2(t)    \right| }{\left| \varphi_{\mathbf{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}_{\mathbf{Y}}^2(t)    \right|  & \geq  \left|   {\varphi}_{\mathbf{Y}}^2(t)    \right| -  \left| \widehat{\varphi}_{\mathbf{Y}}^2(t)  - \varphi_{\mathbf{Y}}^2(t)    \right|  \geq \frac{1}{2} \left|   {\varphi}_{\mathbf{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(  \|  \big(\log  \varphi_{F_0,\sigma_0^2} \big)''(t' \mathbf{A}_k) - \big(\log  \varphi_{P,\Sigma} \big)''(t' \mathbf{A}_k)         \|_{ \mathbb{B}(T_n) }^2 \leq M^2 n^{2 \gamma}  \epsilon_n^2, \| \sigma^{-2} \| \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 $1$.

Recall that we use the posterior measure
\begin{align} \label{qb-posterior-demean-proof-2}  \overline{\nu}_{\alpha,G}(. \: | \: T,k, \mathcal{Z}_n) \sim  Z - \E[Z] \; \; \; \; \; \; \text{where} \; \; \; \; \;  Z  \sim \nu_{\alpha,G}(.|T,k,\mathcal{Z}_n) . \end{align}

Denote the characteristic function of a demeaned Gaussian mixture $\varphi_{P,\sigma^2}$ by $ \overline{\varphi}_{P,\sigma^2} $. For any distribution $Z$, we have $(\log \varphi_{Z} )'' = (\log \varphi_{Z - \E[Z]} )''$.  From this observation and the preceding inequalities for $\nu_{\alpha,G}$, it follows that
\begin{align*}
& \overline{\nu}_{\alpha,G} \bigg(  \|  \big(\log  \varphi_{F_0,\sigma_0^2} \big)''(t' \mathbf{A}_k) - \big(\log  \overline{\varphi}_{P,\sigma^2} \big)''(t' \mathbf{A}_k)         \|_{ \mathbb{B}(T_n) }^2 \leq M^2 n^{2 \gamma}  \epsilon_n^2, \left| \sigma^{-2} \right| \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 $1$.

Denote the elements of $\mathbf{A}_k$ by   $\mathbf{A}_k = (a_1,\dots,a_L)$. Fix any $i$ such that $a_i \neq 0$. Without loss of generality, let $i=1$ and $a_i > 0$. Consider the change of variables
\begin{align*}
z_1 = t' \mathbf{A}_k \; , \; z_2 = t_2 \; , \dots , z_L = t_L.
\end{align*}

The Jacobian of the change of variables $(t_1,\dots,t_L) \rightarrow (z_1,\dots,z_L)$ is given by $ J(z_1,\dots,z_L) =   a_1^{-1}$. Let  $c_L = \inf_{t \in \mathbb{B}(T_n)} t' \mathbf{A}_k $ and $c_U = \sup_{t \in \mathbb{B}(T_n)}  t' \mathbf{A}_k$. It follows that for any non-negative Borel function $f : \R \rightarrow \R_{+}$, we have that
\begin{align*}
\int_{\mathbb{B}(T_n)} f(t' \mathbf{A}_k) dt = \left| a_1 \right|^{-1} (2 T_n)^{d-1}  \int_{c_L}^{c_U} f(z_1)  dz_1.
\end{align*}

In particular, since $ c_U \geq  a_1 T_n $ and $c_L \leq - a_1 T_n $, we have $ \| f   \|_{\mathbb{B}(  a_1  T_n)}^2 \leq D T_n^{1-d}  \| f(t' \mathbf{A}_k)  \|_{\mathbb{B}(T_n)}^2  $ for some universal constant $D > 0$. It follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \overline{\nu}_{\alpha,G} \bigg(  \|  \big(\log  \varphi_{F_0,\sigma_0^2} \big)''(.) - \big(\log  \overline{\varphi}_{P,\sigma^2} \big)''(.)         \|_{ \mathbb{B}( a_1 T_n) }^2 \leq M^2 n^{2 \gamma} T_n^{1-d} \epsilon_n^2, \left| \sigma^{-2} \right| \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 the Gaussian mixture and the true latent distribution are demeaned, we have $ (\log \varphi_{F_0,\sigma_0^2} )'(0) = (\log  \overline{\varphi}_{P,\sigma^2} )'(0) = 0  $. From the Fundemental theorem of calculus and Cauchy-Schwarz, we obtain

\begin{align*}
\left| (\log \varphi_{F_0,\sigma_0^2})' (t) - (\log \overline{\varphi}_{P,\sigma^2})'(t)           \right| & = \left| \int_0^t  \big[ \big(\log  \varphi_{F_0,\sigma_0^2} \big)''(s) - (\log  \overline{\varphi}_{P,\sigma^2} )''(s)     ]  ds  \right| \\ & \leq \sqrt{a_1}  \sqrt{T_n} \|  \big(\log  \varphi_{F_0,\sigma_0^2} \big)''(.) - \big(\log  \overline{\varphi}_{P,\sigma^2} \big)''(.)         \|_{ \mathbb{B}( a_1 T_n) }
\end{align*}

for every $t \in \mathbb{B}(a_1 T_n)$. As all characteristic functions satisfy $\log \varphi (0) = 0 $, we similarly obtain
\begin{align*}
\left| \log \varphi_{F_0,\sigma_0^2} (t) - \log \overline{\varphi}_{P,\sigma^2}(t)           \right| & = \left| \int_0^t  \big[  ( \log \varphi_{F_0,\sigma_0^2})' (s) - ( \log \overline{\varphi}_{P,\Sigma})' (s)    ]  ds  \right| \\ & \leq  \sqrt{a_1}\sqrt{T_n} \| \big(\log  \varphi_{F_0,\sigma_0^2} \big)'(.) - \big(\log  \overline{\varphi}_{P,\sigma^2} \big)'(.)     \|_{\mathbb{B}(a_1T_n)}
\end{align*}

for every $t \in \mathbb{B}(a_1 T_n)$. Furthermore, for every fixed $t \in \R$, the mean value theorem implies that
\begin{align*}
\left|  \varphi_{F_0,\sigma_0^2}(t) -   \overline{\varphi}_{P,\Sigma}(t)   \right|  \leq \sup_{s_t \in [0,1]}  \left| e^{s_t  \log  \varphi_{F_0,\sigma_0^2}(t) + (1-s_t) \log \overline{\varphi}_{P,\sigma^2}(t)  } \right| \left|  \log \varphi_{F_0,\sigma_0^2} (t) - \log \overline{\varphi}_{P,\sigma^2}(t)    \right|.
\end{align*}

Since $ \left| \varphi_{X} \right| \leq 1 $ and $\left| \varphi_{P,\sigma^2}   \right| \leq 1$ (as they are characteristic function of random variables), the preceding bound reduces to
\begin{align*}
\left|  \varphi_{F_0,\sigma_0^2}(t) -   \overline{\varphi}_{P,\sigma^2}(t)   \right|  \leq  \left|  \log \varphi_{F_0,\sigma_0^2} (t) - \log \overline{\varphi}_{P,\sigma^2}(t)    \right|.
\end{align*}

From combining all the preceding bounds, it follows that there exists a universal constant $D > 0$ such that
\begin{align*}
\|   \varphi_{F_0,\sigma_0^2} -  \overline{\varphi}_{P,\sigma^2}  \|_{\mathbb{B}(a_1 T_n)} \leq D T_n^2 \|  \big(\log  \varphi_{F_0,\sigma_0^2} \big)''(.) - \big(\log  \overline{\varphi}_{P,\sigma^2} \big)''(.)  \|_{\mathbb{B}(a_1 T_n)} .
\end{align*}

It follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \overline{\nu}_{\alpha,G} \bigg(  \| \varphi_{F_0,\sigma_0^2}  - \overline{\varphi}_{P,\sigma^2}          \|_{ \mathbb{B}( a_1 T_n) }^2 \leq M^2 n^{2 \gamma} T_n^{5-d} \epsilon_n^2, \left| \sigma^{-2} \right| \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 $\mathbb{B}(a_1T_n)$. Suppose $ \left|  \sigma^{-2}  \right| \leq M^2  \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}  $ holds. It follows that there exists a $c > 0$ for which $  \sigma^2 \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_0^2}   -   \overline{\varphi}_{P,\sigma^2} \big) \mathbbm{1} \{  \left| t \right| >  a_1 T_n  \}   \|_{L^2}^2 & \leq  2 \|  {\varphi}_{F_0,\sigma_0^2}    \mathbbm{1} \{  \left| t \right| >  a_1 T_n  \}   \|_{L^2}^2 +  2 \|     \overline{\varphi}_{P,\sigma^2}  \mathbbm{1} \{ \| t \|_{\infty} > a_1 T_n  \}   \|_{L^2}^2 \\ & \leq 2 \int_{ \left| t \right| > a_1 T_n}  e^{- t^2 \sigma_0^2 } dt + 2 \int_{ \left| t \right| >  a_1 T_n} e^{- t^2 \sigma^2 } dt \\ & \leq 2 \int_{ \left| t \right| > a_1 T_n}  e^{- t^2 \sigma_0^2 } dt + 2 \int_{ \left| t \right| > a_1 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,  $\sigma_0^2$).

With constant $K$ as specified above, it follows that for any $C' > 0$, there exists $ M > 0$ such that
$$ \nu_{\alpha,G} \bigg(  \| \varphi_{F_0,\sigma_0^2}  -  \varphi_{P,\sigma^2}      \|_{L^2} \leq M T_n^{(5-d)/2} n^{\gamma} \epsilon_n  + n^{-K}  \:     \big|\mathcal{Z}_n,T_n  \bigg) \geq 1- \exp(-C' n \epsilon_n^2)  $$

holds with $\mathbb{P}$ probability approaching $1$. The claim follows from observing that
$$ T_n^{(5-d)/2} n^{\gamma} \epsilon_n \asymp n^{-1/2 + \gamma} (\log n)^{\lambda/2 + (5-d)/4 }.  $$

The proof proceeds through several steps which we outline below. We use $D,D' > 0$ as a generic universal constant that may change from line to line. Define
\begin{align*}
\alpha_n^2 = \epsilon_n^{\frac{2}{s+ \zeta}}        \; \; , \; \; \epsilon_n^2 = \frac{(\log n)^{\lambda + d/2}}{n^{\frac{2(s+ 2\zeta) }{2(s+ 2\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{mflb2}  \int   \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T) }^2        \bigg)  d \nu_{\alpha,G} (P,\sigma^2) \geq \exp \big(   - C n    \epsilon_{n}^2   \big)
\end{align}

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

Note that $\left|t' \mathbf{A}_k \right| \leq \sqrt{d} \| t \|_{\infty} \| \mathbf{A}_k \| \leq \sqrt{d}T_n \| \mathbf{A}_k \|   $ uniformly over $t \in \mathbb{B}(T_n)$. Since the preceding constant is finite, it suffices to work under the setting where $\left| t' \mathbf{A}_k \right| \leq D' T_n$.

Observe that the $(l,k)$ element of $  \widehat{\varphi}_{\mathbf{Y}}^2 (t)    \big[ \nabla \nabla' \log \widehat{ \varphi}_{\mathbf{Y}} (t)          \big]$ is given by
$$   -   \widehat{\varphi}_{\mathbf{Y}}(t)     \E_n[Y_l Y_k e^{\mathbf{i} t' \mathbf{Y}}  ] + \E_n[Y_l e^{\mathbf{i} t' \mathbf{Y}}  ] \E_n[ Y_k e^{\mathbf{i} t' \mathbf{Y}}  ]  .   $$

From this representation and an application of Lemma <ref> and <ref>, it follows that
\begin{align*}
\|  \widehat{\varphi}_{\mathbf{Y}}^2 \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}}-  {\varphi}_{\mathbf{Y}}^2 \mathbf{Q}_k^* {\mathcal{V}}_{\mathbf{Y}}     \|_{\mathbb{B}(T)}^2 \leq D \frac{T_n^d \log(T_n)}{n} \leq D \epsilon_n^2
\end{align*}

with $\mathbb{P}$ probability approaching $1$. On this set, it follows that
\begin{align*}
&  \int  \exp \bigg( - \frac{n}{2} \|  \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} -   (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k) \big)    \|_{\mathbb{B}(T_n)}^2       \bigg)  d \nu_{\alpha,G} (P,\sigma^2) \\ & \geq \exp \big(  -nD \epsilon_n^2 \big) \int  \exp \bigg( - nD    \|  {\varphi}_{\mathbf{Y}}^2  \mathbf{Q}_k^* {\mathcal{V}}_{\mathbf{Y}} - \widehat{\varphi}_{\mathbf{Y}}^2  (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)      \|_{\mathbb{B}(T_n)}^2     \bigg)  d \nu_{\alpha,G} (P,\sigma^2).
\end{align*}

From the model, we have that $ \mathbf{Q}_k^* \mathcal{V}_{\mathbf{Y}}(t) = (\log \varphi_{X_k})'' (t'\mathbf{A}_k)   $. By Condition <ref>, there exists universal constants $ \chi , C, M    < \infty$ and a mixing distribution $S_{\alpha_n}$ supported on  $ I_{n} = [-C  (\log \epsilon_n^{-1})^{1/\chi},C  (\log \epsilon_n^{-1})^{1/\chi} ]$ that satisfies
$$ \| \varphi_{\mathbf{Y}}^2 (t) \big[  \big(\log \varphi_{S_{\alpha_n},\alpha_n^2} \big)''(t' \mathbf{A}_k)  - \big(\log \varphi_{X_k} \big)''(t' \mathbf{A}_k)  \big]  \|_{ \mathbb{B}(T) }^2 \leq D \alpha_n^{4 \zeta + 2s}  \leq D \epsilon_n^2 .  $$

It follows that
\begin{align*}
& \int  \exp \bigg( - nD    \|  {\varphi}_{\mathbf{Y}}^2  \mathbf{Q}_k^* {\mathcal{V}}_{\mathbf{Y}} - \widehat{\varphi}_{\mathbf{Y}}^2  (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)      \|_{\mathbb{B}(T_n)}^2     \bigg)  d \nu_{\alpha,G} (P,\sigma^2) \\ &  \geq \exp(-n D \epsilon_n^2) \int  \exp \bigg( - nD    \|  {\varphi}_{\mathbf{Y}}^2 \big(\log \varphi_{S_{\alpha_n},\alpha_n^2} \big)''(t' \mathbf{A}_k)    - \widehat{\varphi}_{\mathbf{Y}}^2  (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)      \|_{\mathbb{B}(T_n)}^2     \bigg)  d \nu_{\alpha,G} (P,\sigma^2) .
\end{align*}

By Condition <ref> and the definition of $(\alpha_n,T_n)$, there exists a $C_1 >0$ such that
$$ \inf_{ \left| t \right| \leq D' T_n }  \left| \varphi_{S_{\alpha_n}}(t) \right| \geq n^{-C_1} .$$

Next, we show that $ S_{\alpha_n} $ can be suitably approximated by a discrete measure. 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_{ \left| t  \right| \leq D' T_n } \left| \partial_{t}^r  \varphi_{S_{\alpha_n}}(t) - \partial_{t}^r \varphi_{F'}(t)  \right| \leq    n^{-2C_1} \epsilon_n   \; \; \; \; \; \; \; r=0,1,2.
\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 $L_2 > 0$. From the preceding bounds, we obtain
\begin{align*}   \sup_{ \left| t \right| \leq D' T_n } \sup_{r=0,1,2}
\frac{\left| \partial_{t}^r  \varphi_{S_{\alpha_n}}(t) - \partial_{t}^r \varphi_{F'}(t)  \right| }{\left| \varphi_{S_{\alpha_n}}(t) \right|^2} \leq D \epsilon_n.
\end{align*}

Since $\epsilon_n \downarrow 0$, observe that this 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 $ \left| t \right| \leq D' T_n$. We write the difference $(\log \varphi_{S_{\alpha_n}})'' -  (\log \varphi_{ F'})'' $ as
\begin{align*}
&  (\log \varphi_{S_{\alpha_n}})'' -  (\log \varphi_{ F'})''  \\ & = (\log \varphi_{S_{\alpha_n}})'' \frac{( \varphi_{F'}^2 - \varphi_{S_{\alpha_n}}^2    )}{ \varphi_{F'}^2  } +  \frac{  \varphi_{S_{\alpha_n}} (  \varphi_{S_{\alpha_n}}''  - \varphi_{F'}''    )      }{ \varphi_{F'}^2 }  +   \frac{  \varphi_{F'}'' (  \varphi_{S_{\alpha_n}} - \varphi_{F'}    )        }{ \varphi_{F'}^2 }        + \frac{ (\varphi_{F'}')^2  - (\varphi_{S_{\alpha_n}}')^2   }{\varphi_{F'}^2}.
\end{align*}

By Condition <ref>, we have $  \left|  (\log \varphi_{ S_{\alpha_n}})''  \right|  \leq D $ and $ \sup_{r=1,2} \left| \partial_{t}^r \varphi_{S_{\alpha_n}}  \right| \leq D$. From the preceding representation and bounds, we then obtain

\begin{align*}
\sup_{ \left| t \right| \leq D' T_n} \left| (\log \varphi_{S_{\alpha_n}})''(t)  -  (\log \varphi_{ F'})'' (t) \right| \leq D \epsilon_n.
\end{align*}

Fix any $L_3 > L_2$  sufficiently large such that
$$   n^{2 C_1 + 1} \epsilon_n^2 \log(n) \epsilon_n^{L_3/2} \leq \epsilon_n \:,\: n^{2 C_1} (\log \epsilon_n)^{2/ \chi} T_n \epsilon_n^{L_3 }  \leq \epsilon_n .  $$

Define $V_i = \{ t \in I_{n} :  \left| t - \mu_i  \right| \leq  \epsilon_n^{L_3}  \} $ for $i=1,\dots,N$ and set $V_0 = \R \setminus \bigcup_{i=1}^N V_i$. 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 $\R$. By Lemma <ref>, for any distribution $P$ that satisfies $  (\int_{\R} \left| x \right|^4 dP(x))^{1/4} \leq \sqrt{n} \epsilon_n \sqrt{\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_{\left| t \right| \leq D' T_n}  \left| \partial_{t}^r \varphi_{F'}(t) - \partial_{t}^r \varphi_{P}(t)    \right| \leq n^{-2 C_1} \epsilon_n \; \; \; \; \; \; \;  r=0,1,2.
\end{align*}

For all such $P$, the preceding bound and an analogous expression to that of $(\log \varphi_{S_{\alpha_n}})''  -  (\log \varphi_{ F'})''$, yields
\begin{align*}
\sup_{ \left| t \right| \leq  D' T_n}  \left| ( \log  \varphi_{ F' } )''  (t) - ( \log \varphi_{P})''(t)     \right| \leq D \epsilon_n.
\end{align*}

From combining all the preceding bounds, observe that all such $P$  also satisfy
\begin{align*} &
\sup_{ \left| t \right| \leq D' T_n}  \left| ( \log  \varphi_{ P } )'' (t) -  (\log \varphi_{S_{\alpha_n}} )''(t)     \right| \leq D \epsilon_n
\\ &
\sup_{\left| t \right| \leq  D' T_n } \left|  ( \log \varphi_P )'' (t)   \right|  \leq D \epsilon_n + \sup_{ \left| t \right| \leq D' T_n } \left| ( \log \varphi_{S_{\alpha_n}})''(t)  \right| \leq D .
\end{align*}

Given any $\sigma^2  \leq D$, the preceding bound also implies that
\begin{align*}
\sup_{ \left| t \right| \leq D'  T_n } \left|  (\log \varphi_{P,\sigma^2})''(t)    \right| &  \leq \sup_{ \left| t \right| \leq D' T_n } \bigg[ \left| ( \log \varphi_{P})''(t)   \right| + \left|  (\log \varphi_{\sigma^2} )''(t)   \right|   \bigg] \\ & \leq D  \big( 1 +  \sigma^2 \big) \\ & \leq D.
\end{align*}

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

holds with $\mathbb{P}$ probability approaching $1$. On this set, when $(P,\sigma^2)$ satisfy the preceding requirements, we have that
\begin{align*}
\|  \widehat{\varphi}_{\mathbf{Y}}^2 ( \log \varphi_{P,\sigma^2})''(t' \mathbf{A}_k)  - {\varphi}_{\mathbf{Y}}^2 ( \log \varphi_{P,\sigma^2})''(t' \mathbf{A}_k)   \|_{\mathbb{B}(T_n)}^2  \leq D \epsilon_n^2.
\end{align*}

Define the sets

\begin{align*}
\Omega_n =  \bigg \{ \sigma^2  \in \R_{+}  : \sigma^2 \in \bigg[  \frac{ \alpha_n^2 }{1 + \epsilon_n  }  ,      \alpha_n^2   \bigg ]    \bigg \}.
\end{align*}

$$ \mathcal{G}_n = \bigg \{ (P,\sigma^2) : \sigma^2 \in \Omega_n , \int_{\R} \left|x \right|^4 dP(x) \leq n^2 \epsilon_n^4 (\log n)^2 \: ,   \sum_{j=1}^N \left| P(V_j) - p_j   \right| \leq \epsilon_n^{L_3}  \bigg  \}.  $$

From combining the preceding bounds, it follows that
\begin{align*}
&  \int  \exp \bigg( - nD    \|  {\varphi}_{\mathbf{Y}}^2 \big(\log \varphi_{S_{\alpha_n},\alpha_n^2} \big)''(t' \mathbf{A}_k)    - \widehat{\varphi}_{\mathbf{Y}}^2  (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)      \|_{\mathbb{B}(T_n)}^2     \bigg)  d \nu_{\alpha,G} (P,\sigma^2) \\ & \geq  \int_{\mathcal{G}_n}  \exp \bigg( - nD    \|  {\varphi}_{\mathbf{Y}}^2 \big(\log \varphi_{S_{\alpha_n},\alpha_n^2} \big)''(t' \mathbf{A}_k)    - \widehat{\varphi}_{\mathbf{Y}}^2  (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)      \|_{\mathbb{B}(T_n)}^2     \bigg)  d \nu_{\alpha,G} (P,\sigma^2) \\ & \geq \exp \big(  -nD \epsilon_n^2 \big) \int_{\mathcal{G}_n}  \exp \bigg( - nD    \|  {\varphi}_{\mathbf{Y}}^2 \big[ \big(\log \varphi_{S_{\alpha_n},\alpha_n^2} \big)''(t' \mathbf{A}_k)    -   (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big]     \|_{\mathbb{B}(T_n)}^2     \bigg)  d \nu_{\alpha,G} (P,\sigma^2).
\end{align*}

For every $(P,\sigma^2) \in \mathcal{G}_n$, the preceding bounds also imply that
\begin{align*}
& \sup_{ \left| t \right| \leq D' T_n}  \left| ( \log  \varphi_{ P , \sigma^2 } )'' (t) -  (\log \varphi_{S_{\alpha_n},\alpha_n^2} )''(t)     \right|  \\ & \leq  \sup_{ \left| t \right| \leq D' T_n}  \left|  ( \log  \varphi_{ P  } )'' (t)  -  (\log \varphi_{S_{\alpha_n}} )''(t)   \right| + \sup_{ \left| t \right| \leq D' T_n}  \left|  ( \log  \varphi_{ \sigma^2  } )'' (t)  -  (\log \varphi_{ \alpha_n^2} )''(t)   \right| \\ & \leq D \epsilon_n + D   \left| \sigma^2 - \alpha_n^2   \right| \\ & \leq D \epsilon_n.
\end{align*}

Since $ \| \varphi_{\mathbf{Y}}^2  \|_{L^2} < \infty$, we obtain that
\begin{align*}
& \int_{\mathcal{G}_n}  \exp \bigg( - nD    \|  {\varphi}_{\mathbf{Y}}^2 \big[ \big(\log \varphi_{S_{\alpha_n},\alpha_n^2} \big)''(t' \mathbf{A}_k)    -   (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big]     \|_{\mathbb{B}(T_n)}^2     \bigg)  d \nu_{\alpha,G} (P,\sigma^2) \\ & \geq \exp \big(  -nD \epsilon_n^2 \big) \int_{ \mathcal{G}_n } d \nu_{\alpha,G} (P,\sigma^2).
\end{align*}

Define the sets
\begin{align*}
&  \mathcal{G}_{n,1} = \bigg \{ (P, \sigma^2) : \sigma^2 \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^2) : \sigma^2 \in \Omega_n , \int_{\R} \left|x \right|^4 dP(x) >  n^2 \epsilon_n^4 (\log n)^2 \: ,   \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} $. Hence
\begin{align*}
\int_{ \mathcal{G}_n }  d \nu_{\alpha,G} (P,\sigma^2) = \int_{ \mathcal{G}_{n,1} }  d \nu_{\alpha,G} (P,\sigma^2)  - \int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G} (P,\sigma^2) .
\end{align*}

For the second term, observe that
\begin{align*}
\int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G} (P,\sigma^2) &  \leq \int_{ P : \int_{\R} \left| x \right|^4 dP(x) >  n^2 \epsilon_n^4 (\log n)^2 }  d \nu_{\alpha,G} (P,\sigma^2) \\ & \leq  \int_{ P : \int_{\R} \left| x \right|^4 dP(x) > n^2 \epsilon_n^4 (\log n)^2  }  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^2) = \int_{  \sigma^2 \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^2).
\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^{L_3}    } 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. The law of $G= G_n$ is given by $\Omega / \sigma_n^2$ where $\Omega \sim L$ and $L$ is a probability measure on  $\R_{+}$ 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^2 \in \Omega_n } d G(\sigma^2) = \int_{  \sigma^2 \in \sigma_n^2 \Omega_n } d L(\sigma^2) \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^2) & = \int_{ \mathcal{G}_{n,1} }  d \nu_{\alpha,G} (P,\sigma^2)  - \int_{ \mathcal{G}_{n,2} }  d \nu_{\alpha,G} (P,\sigma^2)  \\ & \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^2) : \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T_n) }^2 > 2E \epsilon_n^2 }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \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^2 : \left| \sigma^{-2}  \right| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\sigma^2) = \int_{\sigma^2 : \left| \sigma^{-2}  \right| > E (n \epsilon_n^2)^{1/ \kappa}  }   d L(\sigma^2) \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}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T_n) }^2  \geq 0$, the preceding bound implies
\begin{align*}
&  \int_{\sigma^2 : \left| \sigma^{-2}  \right| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }  \exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T_n) }^2        \bigg) d \nu_{\alpha,G} (P,\sigma^2)  \\ & \leq  \int_{\sigma^2 : \left| \sigma^{-2}  \right| > E \sigma_n^2 (n \epsilon_n^2)^{1/ \kappa}  }   d G(\sigma^2)  \\ & \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^2) : \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T_n) }^2 \leq E \epsilon_n^2 , \\ \left| \sigma^{-2} \right| \leq E \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}}}
\exp \bigg( - \frac{n}{2} \| \widehat{\varphi}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \big)      \|_{ \mathbb{B}(T_n) }^2        \bigg)    d \nu_{\alpha,G} (P,\sigma^2) \\ & \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}_{\mathbf{Y}}^2 \big(  \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}} - (\log \varphi_{P,\sigma^2})'' (t'\mathbf{A}_k)  \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 $1$.

The $(l,k)$ element of $  \widehat{\varphi}_{\mathbf{Y}}^2 (t)  \widehat{\mathcal{V}}_{\mathbf{Y}} (t) $ is given by
$$   -   \widehat{\varphi}_{\mathbf{Y}}(t)     \E_n[Y_l Y_k e^{\mathbf{i} t' \mathbf{Y}}  ] + \E_n[Y_l e^{\mathbf{i} t' \mathbf{Y}}  ] \E_n[ Y_k e^{\mathbf{i} t' \mathbf{Y}}  ]  .   $$

From this representation and an application of Lemma <ref> and <ref>, we have that
\begin{align*}
\|  \widehat{\varphi}_{\mathbf{Y}}^2 \mathbf{Q}_k^* \widehat{\mathcal{V}}_{\mathbf{Y}}-  {\varphi}_{\mathbf{Y}}^2 \mathbf{Q}_k^* {\mathcal{V}}_{\mathbf{Y}}     \|_{\mathbb{B}(T)} \leq D \epsilon_n.
\end{align*}

Since $ \mathbf{Q}_k^* \mathcal{V}_{\mathbf{Y}}(t) = (\log \varphi_{X_k})'' (t'\mathbf{A}_k)   $  and  $   \left|  \big( \log  \varphi_{X_k} \big)''(t' \mathbf{A}_k)  \right| \leq D  $, an application of Lemma <ref> implies that
\begin{align*}
\|   \widehat{\varphi}_{\mathbf{Y}}^2  \big( \log  \varphi_{X} \big)''(t' \mathbf{A}_k)  - {\varphi}_{\mathbf{Y}}^2 \mathbf{Q}_k^* {\mathcal{V}}_{\mathbf{Y}}  \|_{\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}_{\mathbf{Y}}^2 \big[ \big(\log  \varphi_{X_k} \big)''(t' \mathbf{A}_k) - \big(\log  \varphi_{P,\sigma^2} \big)''(t' \mathbf{A}_k) \big]         \|_{ \mathbb{B}(T_n) }^2 \leq M^2  \epsilon_n^2, \left| \sigma^{-2} \right| \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}_{\mathbf{Y}}^2(t)  - \varphi_{\mathbf{Y}}^2(t)    \right| }{\left| \varphi_{\mathbf{Y}}^2(t) \right|} \leq  D T_n^{2 \zeta} \frac{\sqrt{\log n}}{\sqrt{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}_{\mathbf{Y}}^2(t)    \right|  & \geq  \left|   {\varphi}_{\mathbf{Y}}^2(t)    \right| -  \left| \widehat{\varphi}_{\mathbf{Y}}^2(t)  - \varphi_{\mathbf{Y}}^2(t)    \right|  \geq \frac{1}{2} \left|   {\varphi}_{\mathbf{Y}}^2(t)   \right|
\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(  \|  \big(\log  \varphi_{X_k} \big)''(t' \mathbf{A}_k) - \big(\log  \varphi_{P,\Sigma} \big)''(t' \mathbf{A}_k)         \|_{ \mathbb{B}(T_n) }^2 \leq M^2 T_n^{4 \zeta}  \epsilon_n^2, \| \sigma^{-2} \| \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 $1$.

Recall that we use the posterior measure
\begin{align} \label{qb-posterior-demean-proof}  \overline{\nu}_{\alpha,G}(. \: | \: T,k, \mathcal{Z}_n) \sim  Z - \E[Z] \; \; \; \; \; \; \text{where} \; \; \; \; \;  Z  \sim \nu_{\alpha,G}(.|T,k,\mathcal{Z}_n) . \end{align}

Denote the characteristic function of a demeaned Gaussian mixture $\varphi_{P,\sigma^2}$ by $ \overline{\varphi}_{P,\sigma^2} $. For any distribution $Z$, we have $(\log \varphi_{Z} )'' = (\log \varphi_{Z - \E[Z]} )''$.  From this observation and the preceding inequalities for $\nu_{\alpha,G}$, it follows that
\begin{align*}
& \overline{\nu}_{\alpha,G} \bigg(  \|  \big(\log  \varphi_{X_k} \big)''(t' \mathbf{A}_k) - \big(\log  \overline{\varphi}_{P,\sigma^2} \big)''(t' \mathbf{A}_k)         \|_{ \mathbb{B}(T_n) }^2 \leq M^2 T_n^{4 \zeta}  \epsilon_n^2, \left| \sigma^{-2} \right| \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 $1$.

Denote the elements of $\mathbf{A}_k$ by   $\mathbf{A}_k = (a_1,\dots,a_L)$. Fix any $i$ such that $a_i \neq 0$. Without loss of generality, let $i=1$ and $a_i > 0$. Consider the change of variables
\begin{align*}
z_1 = t' \mathbf{A}_k \; , \; z_2 = t_2 \; , \dots , z_L = t_L.
\end{align*}

The Jacobian of the change of variables $(t_1,\dots,t_L) \rightarrow (z_1,\dots,z_L)$ is given by $ J(z_1,\dots,z_L) =   a_1^{-1}$. Let  $c_L = \inf_{t \in \mathbb{B}(T_n)} t' \mathbf{A}_k $ and $c_U = \sup_{t \in \mathbb{B}(T_n)}  t' \mathbf{A}_k$. It follows that for any non-negative Borel function $f : \R \rightarrow \R_{+}$, we have that
\begin{align*}
\int_{\mathbb{B}(T_n)} f(t' \mathbf{A}_k) dt = \left| a_1 \right|^{-1} (2 T_n)^{d-1}  \int_{c_L}^{c_U} f(z_1)  dz_1.
\end{align*}

In particular, since $ c_U \geq  a_1 T_n $ and $c_L \leq - a_1 T_n $, we have $ \| f   \|_{\mathbb{B}(  a_1  T_n)}^2 \leq D T_n^{1-d}  \| f(t' \mathbf{A}_k)  \|_{\mathbb{B}(T_n)}^2  $ for some universal constant $D > 0$. It follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \overline{\nu}_{\alpha,G} \bigg(  \|  \big(\log  \varphi_{X_k} \big)''(.) - \big(\log  \overline{\varphi}_{P,\sigma^2} \big)''(.)         \|_{ \mathbb{B}( a_1 T_n) }^2 \leq M^2 T_n^{4 \zeta} T_n^{1-d} \epsilon_n^2, \left| \sigma^{-2} \right| \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 the Gaussian mixture and the true latent distribution are demeaned, we have $ (\log \varphi_{X_k} )'(0) = (\log  \overline{\varphi}_{P,\sigma^2} )'(0) = 0  $. From the Fundemental theorem of calculus and Cauchy-Schwarz, we obtain

\begin{align*}
\left| (\log \varphi_{X_k})' (t) - (\log \overline{\varphi}_{P,\sigma^2})'(t)           \right| & = \left| \int_0^t  \big[ \big(\log  \varphi_{X_k} \big)''(s) - (\log  \overline{\varphi}_{P,\sigma^2} )''(s)     ]  ds  \right| \\ & \leq \sqrt{a_1}  \sqrt{T_n} \|  \big(\log  \varphi_{X_k} \big)''(.) - \big(\log  \overline{\varphi}_{P,\sigma^2} \big)''(.)         \|_{ \mathbb{B}( a_1 T_n) }
\end{align*}

for every $t \in \mathbb{B}(a_1 T_n)$. As all characteristic functions satisfy $\log \varphi (0) = 0 $, we similarly obtain
\begin{align*}
\left| \log \varphi_{X_k} (t) - \log \overline{\varphi}_{P,\sigma^2}(t)           \right| & = \left| \int_0^t  \big[  ( \log \varphi_{X_k})' (s) - ( \log \overline{\varphi}_{P,\Sigma})' (s)    ]  ds  \right| \\ & \leq  \sqrt{a_1}\sqrt{T_n} \| \big(\log  \varphi_{X_k} \big)'(.) - \big(\log  \overline{\varphi}_{P,\sigma^2} \big)'(.)     \|_{\mathbb{B}(a_1T_n)}
\end{align*}

for every $t \in \mathbb{B}(a_1 T_n)$. Furthermore, for every fixed $t \in \R$, the mean value theorem implies that
\begin{align*}
\left|  \varphi_{X_k}(t) -   \overline{\varphi}_{P,\Sigma}(t)   \right|  \leq \sup_{s_t \in [0,1]}  \left| e^{s_t  \log  \varphi_{X_k}(t) + (1-s_t) \log \overline{\varphi}_{P,\sigma^2}(t)  } \right| \left|  \log \varphi_{X_k} (t) - \log \overline{\varphi}_{P,\sigma^2}(t)    \right|.
\end{align*}

Since $ \left| \varphi_{X_k} \right| \leq 1 $ and $\left| \varphi_{P,\sigma^2}   \right| \leq 1$ (as they are characteristic function of random variables), the preceding bound reduces to
\begin{align*}
\left|  \varphi_{X_k}(t) -   \overline{\varphi}_{P,\sigma^2}(t)   \right|  \leq  \left|  \log \varphi_{X_k} (t) - \log \overline{\varphi}_{P,\sigma^2}(t)    \right|.
\end{align*}

From combining all the preceding bounds, it follows that there exists a universal constant $D > 0$ such that
\begin{align*}
\|   \varphi_{X_k} -  \overline{\varphi}_{P,\sigma^2}  \|_{\mathbb{B}(a_1 T_n)} \leq D T_n^2 \|  \big(\log  \varphi_{X_k} \big)''(.) - \big(\log  \overline{\varphi}_{P,\sigma^2} \big)''(.)  \|_{\mathbb{B}(a_1 T_n)} .
\end{align*}

It follows that for any $C' > 0$, there exists a $ M > 0$ such that
\begin{align*}
& \overline{\nu}_{\alpha,G} \bigg(  \| \varphi_{X_k}  - \overline{\varphi}_{P,\sigma^2}          \|_{ \mathbb{B}( a_1 T_n) }^2 \leq M^2  T_n^{4 \zeta + 5-d} \epsilon_n^2, \left| \sigma^{-2} \right| \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 $\mathbb{B}(a_1T_n)$. Suppose $ \left|  \sigma^{-2}  \right| \leq M^2  \sigma_n^2 (n \epsilon_n^2)^{1/\kappa}  $ holds. It follows that there exists a $c > 0$ for which $  \sigma^2 \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}_{X_k}   -   \overline{\varphi}_{P,\sigma^2} \big) \mathbbm{1} \{  \left| t \right| >  a_1 T_n  \}   \|_{L^2}^2 & \leq  2 \|  {\varphi}_{X_k}    \mathbbm{1} \{  \left| t \right| >  a_1 T_n  \}   \|_{L^2}^2 +  2 \|     \overline{\varphi}_{P,\sigma^2}  \mathbbm{1} \{ \| t \|_{\infty} > a_1 T_n  \}   \|_{L^2}^2 \\ & \leq 2 \int_{ \left| t \right| > a_1 T_n}  \left|  \varphi_{X_k} (t) \right|^2  dt + 2 \int_{ \left| t \right| >  a_1 T_n} e^{- t^2 \sigma^2 } dt \\ & \leq 2 \int_{ \left| t \right| > a_1 T_n} \left| \varphi_{X_k} (t) \right|^2  dt + 2 \int_{ \left| t \right| > a_1 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, it follows that for every $C' > 0$, there exists a $ M > 0$ such that
$$ \nu_{\alpha,G} \bigg(  \| \varphi_{X}  -  \varphi_{P,\Sigma}      \|_{L^2} \leq M \big( T_n^{ 2 \zeta + (5-d)/2 } \epsilon_n + T_n^{-p}   \big)   \:     \bigg|\mathcal{Z}_n,T_n  \bigg) \geq 1- \exp(-C' n \epsilon_n^2) . $$

holds with $\mathbb{P}$ probability approaching $1$. The claim follows from observing that
\begin{align*}
T_n^{ 4 \zeta + (5-d) } \epsilon_n^2 \asymp  \frac{(\log n)^{\lambda + 2 \zeta + 5/2 }}{ n^{\frac{2s +d - 5 }{2(s+ 2\zeta) + d }} }.
\end{align*}