\dot{\chi}^{X}_{P}&=\mu^{(1)}(X)-\mu^{(0)}(X)-\chi(P).\end{split}$

By again replacing $Q$ with $\mathbb{Q}_{n}$, we obtain the conditional
average treatment effect (CATE) [Imbens, 2004]

$\theta(P)=\mathbb{Q}_{n}[\mu^{1}-\mu^{0}]$

and the one-step corrected functional for the CATE is

$\tilde{\theta}=\theta(P)+\tilde{P}[\dot{\chi}^{Y}_{P}].$

As before, the CATE is estimated more efficiently than the ATE since we have
removed the uncertainty in the marginal covariate distribution, reducing the
asymptotic posterior variance from $\|\dot{\chi}_{P}\|_{P_{0}}^{2}$ to
$\|\dot{\chi}^{Y}_{P}\|_{P_{0}}^{2}$. The analysis for the CATE is relatively
straightforward so it is omitted; the details can be easily deduced from
Theorems 4 and 13.

###  D.1 Proof of Proposition 12

It is sufficient to show that

$\sqrt{n}(\chi(P_{0})+\mathbb{P}_{n}[\dot{\chi}^{X}_{P_{0}}]-\theta(P_{0}))=o_{P_{0}}(1)$

since this implies that

$\displaystyle\sqrt{n}(\hat{\chi}_{n}-\theta(P_{0}))$
$\displaystyle=\sqrt{n}(\chi(P_{0})+\mathbb{P}_{n}[\dot{\chi}^{Y}_{P_{0}}+\dot{\chi}^{A}_{P_{0}}]+\mathbb{P}_{n}[\dot{\chi}^{X}_{P_{0}}]-\theta(P_{0}))$
$\displaystyle=\sqrt{n}\mathbb{P}_{n}[\dot{\chi}^{Y}_{P_{0}}+\dot{\chi}^{A}_{P_{0}}]+o_{P_{0}}(1)$

and $\sqrt{n}\mathbb{P}_{n}[\dot{\chi}^{Y}_{P_{0}}+\dot{\chi}^{A}_{P_{0}}]$
converges to the requisite normal distribution by the central limit theorem.

Some straightforward algebra establishes

$\sqrt{n}(\chi(P_{0})+\mathbb{P}_{n}[\dot{\chi}^{X}_{P_{0}}]-\theta(P_{0}))=\left[\frac{\sqrt{n}(P_{0}[A]-\mathbb{Q}_{n}[\pi_{0}(X)])}{P_{0}[A]}\right]\left\\{\chi(P_{0})-\frac{\mathbb{Q}_{n}[\pi_{0}(X)\\{\mu_{0}^{(1)}(X)-\mu_{0}^{(0)}(X)\\}]}{\mathbb{Q}_{n}[\pi_{0}(X)]}\right\\}.$

Noting that $P_{0}[A]=P_{0}[\pi_{0}(X)]$, the central limit theorem implies
that $\sqrt{n}(P_{0}[A]-\mathbb{Q}_{n}[\pi_{0}(X)])$ converges in distribution
to a normal distribution with finite variance. By the weak law of large
numbers,

$\displaystyle\mathbb{Q}_{n}[\pi_{0}(X)\\{\mu_{0}^{(1)}(X)-\mu_{0}^{(0)}(X)\\}]$
$\displaystyle\xrightarrow[]{P_{0}}P_{0}[\pi_{0}(X)\\{\mu_{0}^{(1)}(X)-\mu_{0}^{(0)}(X)\\}]$
$\displaystyle\mathbb{Q}_{n}[\pi_{0}(X)]$
$\displaystyle\xrightarrow[]{P_{0}}P_{0}[\pi_{0}(X)].$

Combining these with Slutsky’s lemma, we deduce that the term in curly
brackets is $o_{P_{0}}(1)$. Using Slutsky’s lemma again, we have that
$\sqrt{n}\\{\chi(P_{0})+\mathbb{P}_{n}[\dot{\chi}^{X}_{P_{0}}]-\theta(P_{0})\\}$
converges in distribution to zero. Since zero is a constant, we also have
convergence in probability. ∎

##  Appendix E Proofs of main results

###  E.1 Proof of Theorem 2

By standard arguments, (e.g. p. 142 of van der Vaart [1998] or Castillo and
Rousseau [2015]) it is sufficient to prove the result for the sequence of
posterior laws conditioned on $P\in H_{n}$, i.e.

$d_{BL}(\mathcal{L}_{\Pi\times\Pi_{BB}}(\sqrt{n}(\tilde{\chi}-\hat{\chi}_{n})\mid
Z^{(n)},P\in
H_{n}),\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}}))\xrightarrow[]{P_{0}}0.$

We decompose $\\{\chi(P)-\chi(P_{0})\\}$ as follows:

$\chi(P)-\chi(P_{0})=-P_{0}[\dot{\chi}_{P}]-r_{2}(P_{0},P),$

so we have

$\begin{split}\sqrt{n}(\tilde{\chi}-\hat{\chi}_{n})&=\sqrt{n}\\{\chi(P)-\chi(P_{0})+\tilde{P}[\dot{\chi}_{P}]-\mathbb{P}_{n}[\dot{\chi}_{P_{0}}]\\}\\\
&=\sqrt{n}\\{-P_{0}[\dot{\chi}_{P}]-r_{2}(P_{0},P)+\tilde{P}[\dot{\chi}_{P}]-\mathbb{P}_{n}[\dot{\chi}_{P_{0}}]\\}.\end{split}$

Next, we add and subtract $\sqrt{n}\mathbb{P}_{n}[\dot{\chi}_{P}]$:

$\sqrt{n}(\tilde{\chi}-\hat{\chi}_{n})=\sqrt{n}\\{(\tilde{P}-\mathbb{P}_{n})[\dot{\chi}_{P}]+(\mathbb{P}_{n}-P_{0})[\dot{\chi}_{P}-\dot{\chi}_{P_{0}}]-r_{2}(P_{0},P)\\}.$

Let
$R_{n}(P)=\mathbb{G}_{n}[\dot{\chi}_{P}-\dot{\chi}_{P_{0}}]-\sqrt{n}r_{2}(P_{0},P)$.
Then the decomposition above implies

$\displaystyle\begin{split}&d_{BL}(\mathcal{L}_{\Pi\times\Pi_{BB}}(\sqrt{n}(\tilde{\chi}-\hat{\chi}_{n})\mid
Z^{(n)},P\in H_{n}),\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}}))\\\
&=d_{BL}(\mathcal{L}_{\Pi\times\Pi_{BB}}(\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\dot{\chi}_{P}]+R_{n}(P)\mid
Z^{(n)},P\in H_{n}),\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}}))\\\
&\leq\frac{\sup_{\lVert f\lVert_{BL}\leq 1}\left|\int 1_{P\in
H_{n}}\left\\{\int
f(\sqrt{n}[\tilde{P}-\mathbb{P}_{n}][\dot{\chi}_{P}])\,d\Pi_{BB}(\tilde{P}\mid
Z^{(n)})-\int
f(w)d\mathcal{N}_{(0,\|\dot{\chi}_{P_{0}}\|_{P_{0}}^{2})}(w)\right\\}d\Pi(P\mid
Z^{(n)})\right|}{\int 1_{P\in H_{n}}d\Pi(P\mid Z^{(n)})}\\\
&+\frac{\sup_{\lVert f\lVert_{BL}\leq 1}\left|\int\int 1_{P\in
H_{n}}\left\\{f(\sqrt{n}[\tilde{P}-\mathbb{P}_{n}][\dot{\chi}_{P}]+R_{n}(P))-f(\sqrt{n}[\tilde{P}-\mathbb{P}_{n}][\dot{\chi}_{P}])\right\\}d\Pi_{BB}(\tilde{P}\mid
Z^{(n)})d\Pi(P\mid Z^{(n)})\right|}{\int 1_{P\in H_{n}}d\Pi(P\mid Z^{(n)})}\\\
&\leq
d_{BL}(\mathcal{L}_{\Pi\times\Pi_{BB}}(\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\dot{\chi}_{P}]\mid
Z^{(n)},P\in H_{n}),\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}}))\\\
&+\frac{\int 1_{P\in H_{n}}\left|R_{n}(P)\right|\Pi(P\mid Z^{(n)})}{\int
1_{P\in H_{n}}d\Pi(P\mid Z^{(n)})}\\\ &\leq
d_{BL}(\mathcal{L}_{\Pi\times\Pi_{BB}}(\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\dot{\chi}_{P}]\mid
Z^{(n)},P\in H_{n}),\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}}))\\\
&+\sup_{P\in
H_{n}}|\mathbb{G}_{n}[\dot{\chi}_{P}-\dot{\chi}_{P_{0}}]|+\sup_{P\in
H_{n}}|\sqrt{n}r_{2}(P_{0},P)|.\end{split}$ (27)

First consider the empirical process term on the last line of (27), which is
immediately seen to be $o_{P_{0}}(1)$ under Assumption 1(c∗)(i). With
Assumption 1(c) instead, let $\mathcal{F}$ be a fixed $P_{0}$-Donsker class
such that $\\{\dot{\chi}_{P}:\,P\in H_{n}\\}$ is contained in $\mathcal{F}$
for all sufficiently large $n$. Let
$\rho_{P_{0}}(f)=\sqrt{P_{0}(f-P_{0}f)^{2}}$ and

$\mathcal{F}_{\delta}=\\{f-g:\,f,g\in\mathcal{F},\rho_{P_{0}}(f-g)<\delta\\}$

for any $\delta>0$. Assumption 1(b) implies that

$\displaystyle\delta_{n}=\sup_{P\in
H_{n}}\rho_{P_{0}}(\dot{\chi}_{P}-\dot{\chi}_{P_{0}})\leq\sup_{P\in
H_{n}}\|\dot{\chi}_{P}-\dot{\chi}_{P_{0}}\|_{P_{0}}\rightarrow 0.$

Then for all sufficiently large $n$, $\\{\dot{\chi}_{P}:\,P\in H_{n}\\}$ is
contained in $\mathcal{F}_{2\delta_{n}}$, so Corollary 2.3.12 of van der Vaart
and Wellner [1996] implies that

$\sup_{P\in
H_{n}}|\mathbb{G}_{n}[\dot{\chi}_{P}-\dot{\chi}_{P_{0}}]|\leq\sup_{f\in\mathcal{F}_{2\delta_{n}}}|\mathbb{G}_{n}[f]|\xrightarrow[]{P_{0}^{*}}0,$

where $P_{0}^{*}$ denotes convergence in outer probability.

The third term on the last line of (27)—involving $r_{2}$—tends to zero in
probability by Assumption 1(a). This leaves the first term:

$\displaystyle\begin{split}&d_{BL}(\mathcal{L}_{\Pi\times\Pi_{BB}}(\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\dot{\chi}_{P}]\mid
Z^{(n)},P\in H_{n}),\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}}))\\\
&=\frac{\sup_{\lVert f\lVert_{BL}\leq 1}\left|\int 1_{P\in H_{n}}\left\\{\int
f(\sqrt{n}[\tilde{P}-\mathbb{P}_{n}][\dot{\chi}_{P}])\,d\Pi_{BB}(\tilde{P}\mid
Z^{(n)})-\int
f(w)d\mathcal{N}_{(0,\|\dot{\chi}_{P_{0}}\|_{P_{0}}^{2})}(w)\right\\}d\Pi(P\mid
Z^{(n)})\right|}{\int 1_{P\in H_{n}}d\Pi(P\mid Z^{(n)})}\\\
&\leq\frac{\sup_{\lVert f\lVert_{BL}\leq 1}\left|\int 1_{P\in
H_{n}}\left\\{\int
f(\sqrt{n}[\tilde{P}-\mathbb{P}_{n}][\dot{\chi}_{P}])\,d\Pi_{BB}(\tilde{P}\mid
Z^{(n)})-\int
f(v)d\mathcal{N}_{(0,\|\dot{\chi}_{P}-P_{0}[\dot{\chi}_{P}]\|_{P_{0}}^{2})}(v)\right\\}d\Pi(P\mid
Z^{(n)})\right|}{\int 1_{P\in H_{n}}d\Pi(P\mid Z^{(n)})}\\\
&+\frac{\sup_{\lVert f\lVert_{BL}\leq 1}\left|\int 1_{P\in H_{n}}\left\\{\int
f(v)d\mathcal{N}_{(0,\|\dot{\chi}_{P}-P_{0}[\dot{\chi}_{P}]\|_{P_{0}}^{2})}(v)-\int
f(w)d\mathcal{N}_{(0,\|\dot{\chi}_{P_{0}}\|_{P_{0}}^{2})}(w)\right\\}d\Pi(P\mid
Z^{(n)})\right|}{\int 1_{P\in H_{n}}d\Pi(P\mid Z^{(n)})}\\\ &\leq\sup_{\lVert
f\lVert_{BL}\leq 1}\sup_{P\in H_{n}}\left|\int
f(\sqrt{n}[\tilde{P}-\mathbb{P}_{n}][\dot{\chi}_{P}])\,d\Pi_{BB}(\tilde{P}\mid
Z^{(n)})-\int
f(v)d\mathcal{N}_{(0,\|\dot{\chi}_{P}-P_{0}[\dot{\chi}_{P}]\|_{P_{0}}^{2})}(v)\right|\\\
&+\sup_{\lVert f\lVert_{BL}\leq 1}\sup_{P\in H_{n}}\left|\int
f(v)d\mathcal{N}_{(0,\|\dot{\chi}_{P}-P_{0}[\dot{\chi}_{P}]\|_{P_{0}}^{2})}(v)-\int
f(w)d\mathcal{N}_{(0,\|\dot{\chi}_{P_{0}}\|_{P_{0}}^{2})}(w)\right|\\\
&=\sup_{P\in
H_{n}}d_{BL}(\mathcal{L}_{\Pi_{BB}}(\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\dot{\chi}_{P}]\mid
Z^{(n)}),\mathcal{N}(0,\|\dot{\chi}_{P}-P_{0}[\dot{\chi}_{P}]\|^{2}_{P_{0}}))\\\
&+\sup_{P\in
H_{n}}d_{BL}(\mathcal{N}(0,\|\dot{\chi}_{P}-P_{0}[\dot{\chi}_{P}]\|^{2}_{P_{0}}),\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}})).\end{split}$
(28)

By Assumption 1(c), we can apply Theorem 3.6.13 of van der Vaart and Wellner
[1996] to show that the first term on the last line of (LABEL:eqn::BB_ineq)
tends to zero in outer probability. Otherwise, we apply Lemma 11 under
Assumption 1(c∗). The second term on the last line of (LABEL:eqn::BB_ineq)
tends to zero via the identity
$d_{BL}(\mathcal{N}(0,\sigma^{2}),\mathcal{N}(0,\tau^{2}))\leq\sqrt{|\sigma^{2}-\tau^{2}|}$
and the assumed uniform $L_{2}$-convergence in Assumption 1(b), which
completes the proof.

If the efficient influence function takes the form

$\dot{\chi}_{P}=\psi_{P}-P[\psi_{P}],$

we point out that

$\displaystyle\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\dot{\chi}_{P}]$
$\displaystyle=\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\psi_{P}]$
$\displaystyle\mathbb{G}_{n}[\dot{\chi}_{P}-\dot{\chi}_{P_{0}}]$
$\displaystyle=\mathbb{G}_{n}[\psi_{P}-\psi_{P_{0}}].$

The proof can then be easily adapted when $\dot{\chi}_{P}$ and
$\dot{\chi}_{P_{0}}$ are replaced by $\psi_{P}$ and $\psi_{P_{0}}$
respectively in Assumptions 1(b), (c) and (c*). ∎

###  E.2 Proof of Theorem 3

From Shen et al. [2013], let
$\tilde{\epsilon}_{n}=M_{n}n^{-\alpha/(2\alpha+1)}$ with $M_{n}\geq(\log
n)^{r}$ and

$\mathcal{F}_{n}=\left\\{f(x)=\int\varphi_{\sigma}(x-\mu)dQ(\mu),Q\in\mathcal{Q}_{n},\sigma\in(\sigma_{n},S_{n})\right\\}\cap\\{d_{H}(f_{0},f)\leq\tilde{\epsilon}_{n}\\},\quad$

for some $r>0$ with

$\mathcal{Q}_{n}=\left\\{\sum_{j}p_{j}\delta_{(\mu_{j})},\sum_{j>J_{n}}p_{j}\leq\sigma_{n}n^{-2},\max_{j\leq
J_{n}}|\mu_{j}|\leq n^{B}\right\\},\quad
J_{n}=J_{0}n\tilde{\epsilon_{n}}^{2},\quad B,J_{0}>0$

and $\sigma_{n}=\sigma_{0}\tilde{\epsilon}_{n}^{1/\alpha}$ and
$S_{n}=e^{s_{0}n\tilde{\epsilon}_{n}^{2}}$ for some $s_{0}>0$. Then from Shen
et al. [2013] Proposition 2 and Theorem 1,

$\Pi(\mathcal{Q}_{n}^{c}|Y^{n})=o_{P_{0}}(1),\quad\Pi(d_{H}(f_{0},f)>\tilde{\epsilon}_{n}|Y^{n})=o_{P_{0}}(1).$

Thus if
$H_{n}=\mathcal{F}_{n}\cap\\{\|f-f_{0}\|_{2}>M_{0}\tilde{\epsilon}_{n}\\}$,
$\Pi(H_{n}^{c}|Y^{n})=o_{P_{0}}(1)$. We write
$\mathcal{G}_{n}=\\{g=f_{J_{n}}-f_{0},f\in H_{n}\\}$, where we denote for
$f(x)=\sum_{j=1}^{\infty}p_{j}\varphi_{\sigma}(x-\mu_{j})$,
$f_{J_{n}}(x)=\sum_{j=1}^{J_{n}}p_{j}\varphi_{\sigma}(x-\mu_{j})$. To prove
assumption 1 we first prove that on $H_{n}$ the densities are bounded and that
$\|g\|_{2}=O(\tilde{\epsilon}_{n})$. First we show that

$\Pi(\|f\|_{\infty}>4\|f_{0}\|_{\infty}|Y^{n})=o_{p_{0}}(1)$ (29)

First, on the Hellinger ball $d_{H}(f,f_{0})\leq\tilde{\epsilon}_{n}$, we have

$\int\mathbf{1}_{f>4\|f_{0}\|_{\infty}}f(x)dx\leq 4d_{H}(f_{0},f)^{2}\leq
4\tilde{\epsilon}_{n}^{2},$

which in terms implies that

$\text{Leb}(\\{f\geq
4\|f_{0}\|_{\infty}\\})\leq\frac{\tilde{\epsilon}_{n}^{2}}{\|f_{0}\|_{\infty}}.$

Let $x_{1}\in A_{n}=\\{x;f(x)>8\|f_{0}\|_{\infty}\\}$, noting that for all
$f\in\mathcal{F}_{n}$,
$|f^{{}^{\prime}}(y)|\lesssim\sigma^{-2}\lesssim\sigma_{n}^{-2},$ we have that

$f(x)\geq f(x_{1})-|x-x_{1}|C/\sigma_{n}^{2}\geq 4\|f_{0}\|_{\infty}$

as soon as $|x-x_{1}|\leq
2\sigma_{n}^{2}/C\lesssim\tilde{\epsilon}_{n}^{2/\alpha}>>\tilde{\epsilon}_{n}^{2}$
when $\alpha>1$, which contradicts
$\text{Leb}(A_{n})\lesssim\tilde{\epsilon_{n}}^{2}$.

Interestingly (29) allows us to refine the $L_{2}$ rate obtained by Scricciolo
[2014] since

$\|f-f_{0}\|_{2}^{2}\leq
2d_{H}(f,f_{0})^{2}(\|f\|_{\infty}+\|f_{0}\|_{\infty})\lesssim\tilde{\epsilon}_{n}^{2}.$

So that we obtain

$\Pi(\|f-f_{0}\|_{2}>M_{0}\tilde{\epsilon}_{n}|Y^{n})=o_{P_{0}}(1).$

while the rate obtained in Scricciolo [2014] was
$\tilde{\epsilon}_{n}(n\tilde{\epsilon}_{n}^{2})^{1/4}$.

Hereafter we write $\dot{\chi}_{f}$ in place of $\dot{\chi}_{P}$. Then, when
$\alpha>1/2$ and uniformly over $H_{n}$,

$\sqrt{n}r_{2}(f_{0},f)=\chi(f_{0})-\chi(f)-P_{0}(\dot{\chi}_{f})=\sqrt{n}\|f-f_{0}\|_{2}^{2}=o(1),$

and

$\|\dot{\chi}_{f}-\dot{\chi}_{f_{0}}\|_{f_{0}}=4\|f-f_{0}\|_{2}^{2}=o(1).$

We now prove Assumption 1 $(c^{*})$, using Lemma 19.36 of van der Vaart
[1998]. For all $g\in\mathcal{G}_{n}$

$F_{0}(g^{2})\leq
2\|f_{0}\|_{\infty}\tilde{\epsilon}_{n}^{2}+2F_{0}((f-f_{J_{n}})^{2})\leq
2\|f_{0}\|_{\infty}\tilde{\epsilon}_{n}^{2}+2n^{-4}/\sigma_{n}^{2}\leq
3\|f_{0}\|_{\infty}\tilde{\epsilon}_{n}^{2}:=\delta_{n}^{2},$

for $n$ large enough and $\|g\|_{\infty}\leq 1/\sigma_{n}$. We thus only need
to upper bound
$J_{[],n}:=J_{[]}(\delta_{n},\mathcal{G}_{n},L_{2}(f_{0}))=o(n^{1/4}\tilde{\epsilon}_{n}\wedge
1)$ so that using $\|g\|_{\infty}\leq 2\|f_{0}\|_{\infty}$

$\mathbb{E}_{0}^{*}\sup_{g\in\mathcal{G}_{n}}\mathbb{G}_{n}(g)\lesssim
J_{[],n}+\frac{J_{[],n}^{2}}{\delta_{n}^{2}\sqrt{n}}=o(1)$

In Lemma 14 we show that

$J_{[],n}\lesssim\sqrt{J_{n}}\tilde{\epsilon}_{n}\log
n\lesssim\sqrt{n}\tilde{\epsilon}_{n}^{2}\log n\asymp
n^{-(\alpha-1/2)/(2\alpha+1)}(\log n)^{r_{1}}$

for some $r_{1}$, so that as soon as $\alpha>1/2$, $J_{[],n}==o(1)$ and
$J_{[],n}^{2}/\tilde{\epsilon}_{n}^{2}\lesssim n\tilde{\epsilon}_{n}^{2}\log
n=o(\sqrt{n})$ for $\alpha>1/2$. Therefore,

$\sup_{g\in\mathcal{G}_{n}}|\mathbb{G}_{n}(g)|=o_{p_{0}}(1).$

Finally on $\mathcal{F}_{n}$ $\|f\|_{\infty}\leq 4\|f_{0}\|_{\infty}$,
therefore an envelop function of $\mathcal{F}_{n}$ is the constant
$4\|f_{0}\|_{\infty}$ and Assumption 1 (ii) is verified which terminates the
proof of Theorem 3. ∎

######  Lemma 14.

Using the definitions above

$J_{[],n}=O(\sqrt{J_{n}}\tilde{\epsilon}_{n}\log n).$

###### Proof of Lemma 14.

Recall that, writing $\delta_{n}=\|f_{0}\|_{\infty}\epsilon_{n,2}$

$J_{[],n}=\int_{0}^{\delta_{n}}\sqrt{\log
N_{[]}(u,\mathcal{G}_{n},L_{2}(f_{0}))}du$

where $N_{[]}(u,\mathcal{G}_{n},L_{2}(f_{0}))$ is the bracketing covering
number . Let $\zeta>0$, let $\mathcal{S}_{\mu}=(\hat{\mu}_{\ell},\ell\leq
L_{1}(u))$ be a net of radius $\zeta u\sigma_{n}^{2}$ of $[-a_{n},a_{n}]$ ,
$\mathcal{S}_{\sigma}=\hat{\sigma}_{\ell}=\sigma_{n}(1+\zeta u^{2})^{\ell}$,
$\ell\leq L_{2}(u)\lesssim\frac{\log(S_{n}/\sigma_{n})}{u^{2}}$ and
$\mathcal{S}_{p}=(\hat{p}^{(\ell)},\ell\leq L_{3}(u))$ a net of $[0,1]$ of
radius $\sigma_{n}u/J_{n}$. Consider the set of functions

$\hat{g}^{U}(x)=\sum_{j=1}^{J_{n}}(\hat{p}_{j}+\sigma_{n}u^{2}/J_{n})(1+u^{2})\varphi_{\hat{\sigma}/(1+u^{2})}(x-\hat{\mu}_{j})^{2}e^{|x-\hat{\mu}_{j}|\zeta
u}-f_{0}$
$\hat{g}^{L}(x)=\sum_{j=1}^{J_{n}}(\hat{p}_{j}-\sigma_{n}u^{2}/J_{n})_{+}(1+u^{2})\varphi_{\hat{\sigma}/(1-u^{2})}(x-\hat{\mu}_{j})^{2}e^{-|x-\hat{\mu}_{j}|\zeta
u}-f_{0}$

Let $g\in\mathcal{G}_{n}$, we show that there exists a pair
$\hat{g}^{U},\hat{g}^{L}$ as above such that $\hat{g}^{U}\geq
g\geq\hat{g}^{L}$. Then since the number of such pairs is bounded by
$L_{1}(u)^{J_{n}}L_{2}(u)L_{3}(u)^{J_{n}}$,

$\displaystyle\log N_{[]}(u,\mathcal{G}_{n},L_{2}(f_{0}))$ $\displaystyle\leq
J_{n}[\log L_{1}(u)+\log L_{3}(u)]+\log L_{2}(u)$ $\displaystyle\lesssim
J_{n}(\log(a_{n}/\sigma_{n})+\log(J_{n}/\sigma_{n})+\log(1/u))+\log\log(S_{n}/\sigma_{n})+\log(1/u).$

By definition
$\log(a_{n}/\sigma_{n})+\log(J_{n}/\sigma_{n})+\log\log(S_{n}/\sigma_{n})\lesssim\log
n$ so that

$J_{[],n}=\sqrt{J_{n}}\sqrt{\log
n}\int_{0}^{\delta_{n}}\sqrt{\log(1/u)}du\lesssim\sqrt{J_{n}}\delta_{n}\log
n.$

∎

###  E.3 Proof of Theorem 4

Let

$\psi_{P}=\chi(P)+\dot{\chi}_{P}=\frac{A}{\pi(X)}\\{Y-m(X)\\}+m(X),$

such that $\dot{\chi}_{P}=\psi_{P}-P[\psi_{P}]$. As we have remarked in the
main paper, it is sufficient to verify Assumption 1 with $\dot{\chi}_{P}$ and
$\dot{\chi}_{P_{0}}$ replaced by $\psi_{P}$ and $\psi_{P_{0}}$ respectively
for (b) and (c). Assumption 1(a) is verified by the inequality in (3) and
Assumption 2(a).

Note that

$\psi_{P}-\psi_{P_{0}}=\left(1-\frac{A}{\pi(X)}\right)[m(X)-m_{0}(X)]+A(Y-m(X))\left[\frac{1}{\pi(X)}-\frac{1}{\pi_{0}(X)}\right],$

so by Assumption 2(b), we have

$\|\psi_{P}-\psi_{P_{0}}\|_{P_{0}}\leq\left(1+\frac{1}{\delta}\right)\|m-m_{0}\|_{P_{0}}+C\left\|\frac{1}{\pi}-\frac{1}{\pi_{0}}\right\|_{P_{0}}.$

We deduce by Assumption 2(a) that

$\sup_{P\in H_{n}}\|\psi_{P}-\psi_{P_{0}}\|_{P_{0}}\rightarrow 0.$ (30)

With the uniform bounding in Assumption 2(b) and the permanence of the Donsker
property under Lipschitz transformations (see examples 2.10.7, 2.10.8 and
2.10.9 in van der Vaart and Wellner [1996]), the classes $\\{\psi_{P}:P\in
H_{n}\\}$ eventually lie in a fixed $P_{0}$-Donsker class. ∎

###  E.4 Proof of Theorem 6

To aid comparisons with Ray and van der Vaart [2020], we use the following
notation:

$\displaystyle a$ $\displaystyle=\frac{1}{\pi},\quad
a_{0}=\frac{1}{\pi_{0}},\quad\hat{a}_{n}=\frac{1}{\hat{\pi}_{n}}$
$\displaystyle b$ $\displaystyle=m,\quad b_{0}=m_{0}.$

Similar to the argument in the proof of Theorem 2 of Ray and van der Vaart
[2020], it suffices to fix $\hat{a}_{n}$ to be a deterministic sequence
$a_{n}\in\mathcal{A}_{n}$, where $\mathcal{A}_{n}=\\{a:\,\|a\|_{\infty}\leq
M,\|a-a_{0}\|_{P_{0}}\leq M\rho_{n}\\}$ for any $M>0$. This is due to the
assumed stochastic independence of the data and $\hat{a}_{n}$, as well as the
assumptions that $\|\hat{a}_{n}\|_{\infty}=O_{P_{0}}(1)$ and
$\|\hat{a}_{n}-a_{0}\|_{P_{0}}=O_{P_{0}}(\rho_{n})$. Like the proof of Theorem
4, we set our sequence of efficient estimators to be

$\hat{\chi}_{n}=\mathbb{P}_{n}[Aa_{0}(X)(Y-b_{0}(X))+b_{0}(X)].$

Then we have

$\displaystyle\sqrt{n}(\tilde{\chi}-\hat{\chi}_{n})=$
$\displaystyle\,\sqrt{n}\tilde{P}[Aa_{n}(X)(Y-b(X))+b(X)]-\sqrt{n}\mathbb{P}_{n}[Aa_{0}(X)(Y-b_{0}(X))+b_{0}(X)]$
$\displaystyle=$
$\displaystyle\underbrace{\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[Aa_{n}(X)(Y-b(X))+b(X)]}_{\leavevmode\hbox
to8.9pt{\vbox to8.9pt{\pgfpicture\makeatletter\hbox{\hskip
4.45195pt\lower-4.45195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.25195pt}{0.0pt}\pgfsys@curveto{4.25195pt}{2.34831pt}{2.34831pt}{4.25195pt}{0.0pt}{4.25195pt}\pgfsys@curveto{-2.34831pt}{4.25195pt}{-4.25195pt}{2.34831pt}{-4.25195pt}{0.0pt}\pgfsys@curveto{-4.25195pt}{-2.34831pt}{-2.34831pt}{-4.25195pt}{0.0pt}{-4.25195pt}\pgfsys@curveto{2.34831pt}{-4.25195pt}{4.25195pt}{-2.34831pt}{4.25195pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle+\underbrace{\mathbb{G}_{n}[Aa_{n}(X)(Y-b(X))+b(X)-Aa_{0}(X)(Y-b_{0}(X))-b_{0}(X)]}_{\leavevmode\hbox
to8.9pt{\vbox to8.9pt{\pgfpicture\makeatletter\hbox{\hskip
4.45195pt\lower-4.45195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.25195pt}{0.0pt}\pgfsys@curveto{4.25195pt}{2.34831pt}{2.34831pt}{4.25195pt}{0.0pt}{4.25195pt}\pgfsys@curveto{-2.34831pt}{4.25195pt}{-4.25195pt}{2.34831pt}{-4.25195pt}{0.0pt}\pgfsys@curveto{-4.25195pt}{-2.34831pt}{-2.34831pt}{-4.25195pt}{0.0pt}{-4.25195pt}\pgfsys@curveto{2.34831pt}{-4.25195pt}{4.25195pt}{-2.34831pt}{4.25195pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle+\underbrace{\sqrt{n}P_{0}[Aa_{n}(X)(Y-b(X))+b(X)-Aa_{0}(X)(Y-b_{0}(X))-b_{0}(X)]}_{\leavevmode\hbox
to8.9pt{\vbox to8.9pt{\pgfpicture\makeatletter\hbox{\hskip
4.45195pt\lower-4.45195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.25195pt}{0.0pt}\pgfsys@curveto{4.25195pt}{2.34831pt}{2.34831pt}{4.25195pt}{0.0pt}{4.25195pt}\pgfsys@curveto{-2.34831pt}{4.25195pt}{-4.25195pt}{2.34831pt}{-4.25195pt}{0.0pt}\pgfsys@curveto{-4.25195pt}{-2.34831pt}{-2.34831pt}{-4.25195pt}{0.0pt}{-4.25195pt}\pgfsys@curveto{2.34831pt}{-4.25195pt}{4.25195pt}{-2.34831pt}{4.25195pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$

Term 3 is just the second-order bias

3
$\displaystyle=\sqrt{n}P_{0}\left[\frac{1}{a_{0}(X)}\left(a_{0}(X)-a_{n}(X)\right)\\{b_{0}(X)-b(X)\\}\right]$
$\displaystyle\leq\sqrt{n}\left\|\frac{1}{a_{0}}\right\|_{\infty}\left\|a_{0}-a_{n}\right\|_{P_{0}}\|b_{0}-b\|_{P_{0}},$

and we can upper bound this by $\sqrt{n}M\rho_{n}\varepsilon_{n}\rightarrow 0$
for $b$ in $\\{b=\Psi(w):w\in\tilde{\mathcal{H}}_{n}\\}$ by (11). For term 2,
we have the inequality

$\displaystyle|\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}|\leq$
$\displaystyle\,|\mathbb{G}_{n}[Aa_{n}(X)(Y-b(X))-Aa_{0}(X)(Y-b_{0}(X))]|+|\mathbb{G}_{n}[b(X)-b_{0}(X)]|.$

The second term on the right converges uniformly in probability to zero on
$\\{b=\Psi(w):w\in\tilde{\mathcal{H}}_{n}\\}$ by (12). We can decompose the
first term on the right as follows:

$\displaystyle|\mathbb{G}_{n}[Aa_{n}(X)(Y-b(X))-Aa_{0}(X)(Y-b_{0}(X))]|\leq$
$\displaystyle\,\underbrace{|\mathbb{G}_{n}[Aa_{n}(X)\\{b_{0}(X)-b(X)\\}]|}_{\leavevmode\hbox
to10.32pt{\vbox to10.32pt{\pgfpicture\makeatletter\hbox{\hskip
5.1624pt\lower-5.1624pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.9624pt}{0.0pt}\pgfsys@curveto{4.9624pt}{2.74069pt}{2.74069pt}{4.9624pt}{0.0pt}{4.9624pt}\pgfsys@curveto{-2.74069pt}{4.9624pt}{-4.9624pt}{2.74069pt}{-4.9624pt}{0.0pt}\pgfsys@curveto{-4.9624pt}{-2.74069pt}{-2.74069pt}{-4.9624pt}{0.0pt}{-4.9624pt}\pgfsys@curveto{2.74069pt}{-4.9624pt}{4.9624pt}{-2.74069pt}{4.9624pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.625pt}{-2.39166pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{A}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle+\underbrace{|\mathbb{G}_{n}[A\\{a_{n}(X)-a_{0}(X)\\}\\{Y-b_{0}(X)\\}]|}_{\leavevmode\hbox
to10.08pt{\vbox to10.08pt{\pgfpicture\makeatletter\hbox{\hskip
5.04057pt\lower-5.04057pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.84058pt}{0.0pt}\pgfsys@curveto{4.84058pt}{2.6734pt}{2.6734pt}{4.84058pt}{0.0pt}{4.84058pt}\pgfsys@curveto{-2.6734pt}{4.84058pt}{-4.84058pt}{2.6734pt}{-4.84058pt}{0.0pt}\pgfsys@curveto{-4.84058pt}{-2.6734pt}{-2.6734pt}{-4.84058pt}{0.0pt}{-4.84058pt}\pgfsys@curveto{2.6734pt}{-4.84058pt}{4.84058pt}{-2.6734pt}{4.84058pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.47917pt}{-2.39166pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{B}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$

We need to show that each term converges uniformly in probability to zero on
$\\{b=\Psi(w):w\in\tilde{\mathcal{H}}_{n}\\}$. For term A, this is achieved by
applying Lemmas 10 and 9 of Ray and van der Vaart [2020] with $\varphi=Aa_{n}$
and $h=b-b_{0}$ with the assumptions of (11) and (12), along with the uniform
bounding for $a_{n}$. Since $a_{n}$ is a deterministic sequence that converges
in $L_{2}(P_{0})$ to $a_{0}$, term B converges to zero in probability.

It remains to establish the convergence of term 1 to the requisite normal
distribution. Having shown that term 2 converges uniformly in probability to
zero on $\\{b=\Psi(w):w\in\tilde{\mathcal{H}}_{n}\\}$, we already have the
requisite Glivenko-Cantelli condition to apply Lemma 11 (the bounding on the
envelope functions is immediately implied by $\|a_{n}\|_{\infty}\leq M$).
Thus, we just need to show that $[Aa_{n}(X)(Y-b(X))+b(X)]$ converges uniformly
in $L_{2}$ to $[Aa_{0}(X)(Y-b_{0}(X))+b_{0}(X)]$. Following the steps in the
proof of Theorem 4, we have

$\|Aa_{n}(X)(Y-b(X))+b(X)-Aa_{0}(X)(Y-b_{0}(X))-b_{0}(X)\|_{P_{0}}\leq\left(1+M\right)\|b-b_{0}\|_{P_{0}}+\left\|a_{n}-a_{0}\right\|_{P_{0}},$

so we are done after applying (11). ∎

###  E.5 Proof of Corollary 7

We prove this corollary by applying several of the results in Ray and van der
Vaart [2020] to check the conditions in Theorem 6. First, by applying Lemma 5
of Ray and van der Vaart [2020] with $t=0$, there exist measurable subsets
$\mathcal{H}_{n}\subset C([0,1])$ (the set of continuous functions on $[0,1]$)
with

$\displaystyle\Pi(w\in\mathcal{H}_{n}\mid Z^{(n)})$
$\displaystyle\xrightarrow[]{P_{0}}1$
$\displaystyle\sup_{m=\Psi(w):w\in\mathcal{H}_{n}}|\mathbb{G}_{n}[m-m_{0}]|$
$\displaystyle\xrightarrow[]{P_{0}}0.$

Next, Lemma 16 and the proof of Corollary 1 for the Riemann-Liouville prior
from Ray and van der Vaart [2020] implies that $m$ contracts to $m_{0}$ at the
rate

$\varepsilon_{n}=n^{-\frac{\beta\wedge\bar{\beta}}{2\bar{\beta}+1}}(\log
n)^{\kappa},$

where $\kappa$ either takes the value 0 or 1 depending on the values of
$(\beta,\bar{\beta})$ (see Theorem 4 of Castillo [2008] or the discussion
following Lemma 11.34 in Ghosal and van der Vaart [2017]). In other words,

$\Pi(\|m-m_{0}\|_{P_{0}}\leq M\varepsilon_{n}\mid
Z^{(n)})\xrightarrow[]{P_{0}}1$

for some sufficiently large constant $M>0$ (see also: Theorem 11.22 of Ghosal
and van der Vaart [2017]). Setting $\kappa$ to be identically equal to one
also suffices. (This was done in the statement of the corollary for the sake
of simplicity.) So the sequence of subsets

$\tilde{\mathcal{H}}_{n}=\mathcal{H}_{n}\cap\\{w\in
C([0,1]):\|\Psi(w)-m_{0}\|_{P_{0}}\leq M\varepsilon_{n}\\}$

satisfies the conditions for Theorem 6. ∎

###  E.6 Proof of Theorem 8

Our sequence of efficient estimators is defined by

$\hat{\chi}_{n}=\frac{\mathbb{P}_{n}\left[\left(\frac{A-\pi_{0}(X)}{1-\pi_{0}(X)}\right)\\{Y-\mu_{0}^{(0)}(X)\\}\right]}{\mathbb{P}_{n}[A]}.$

We begin by showing that this sequence does indeed achieve the requisite
limiting distribution.

$\displaystyle\sqrt{n}(\hat{\chi}_{n}-\chi(P_{0}))$
$\displaystyle=\frac{\sqrt{n}\mathbb{P}_{n}\left[\left(\frac{A-\pi_{0}(X)}{1-\pi_{0}(X)}\right)\\{Y-\mu_{0}^{(0)}(X)\\}-A\chi(P_{0})\right]}{\mathbb{P}_{n}[A]}$
$\displaystyle=\sqrt{n}\mathbb{P}_{n}[\dot{\chi}_{P_{0}}]\frac{P_{0}[A]}{\mathbb{P}_{n}[A]}$
$\displaystyle\rightsquigarrow\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}}),$

since the ratio $P_{0}[A]/\mathbb{P}_{n}[A]$ converges in probability to one
by the weak law of large numbers and Slutsky’s lemma.

Now define

$\displaystyle h(Z)$
$\displaystyle=\left(\frac{A-\pi(X)}{1-\pi(X)}\right)\\{Y-\mu^{(0)}(X)\\}-A\chi(P_{0})$
$\displaystyle h_{0}(Z)$
$\displaystyle=\left(\frac{A-\pi_{0}(X)}{1-\pi_{0}(X)}\right)\\{Y-\mu_{0}^{(0)}(X)\\}-A\chi(P_{0})=P_{0}[A]\dot{\chi}_{P_{0}}(Z).$

With the uniform bounding in Assumption 4(b) and the permanence of the Donsker
property under Lipschitz transformations (see examples 2.10.7, 2.10.8 and
2.10.9 in van der Vaart and Wellner [1996]), the classes $\\{h:P\in H_{n}\\}$
eventually lie in a fixed $P_{0}$-Donsker class. We have

$\displaystyle\sqrt{n}(\tilde{\chi}-\hat{\chi}_{n})$
$\displaystyle=\sqrt{n}\\{\tilde{\chi}-\chi(P_{0})-\hat{\chi}_{n}+\chi(P_{0})\\}$
$\displaystyle=\frac{\sqrt{n}\tilde{P}\left[\left(\frac{A-\pi(X)}{1-\pi(X)}\right)\\{Y-\mu^{(0)}(X)\\}-A\chi(P_{0})\right]}{\tilde{P}[A]}-\frac{\sqrt{n}\mathbb{P}_{n}\left[\left(\frac{A-\pi_{0}(X)}{1-\pi_{0}(X)}\right)\\{Y-\mu_{0}^{(0)}(X)\\}-A\chi(P_{0})\right]}{\mathbb{P}_{n}[A]}$
$\displaystyle=\frac{\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[h(Z)]+\sqrt{n}\mathbb{P}_{n}[h(Z)]}{\tilde{P}[A]}-\frac{\sqrt{n}\mathbb{P}_{n}[h_{0}(Z)]}{\mathbb{P}_{n}[A]}$
$\displaystyle=\underbrace{\frac{\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[h(Z)]}{P_{0}[A]}+\frac{\sqrt{n}(\mathbb{P}_{n}-P_{0})[h(Z)-h_{0}(Z)]}{\mathbb{P}_{n}[A]}+\frac{\sqrt{n}P_{0}[h(Z)]}{\mathbb{P}_{n}[A]}}_{\leavevmode\hbox
to8.9pt{\vbox to8.9pt{\pgfpicture\makeatletter\hbox{\hskip
4.45195pt\lower-4.45195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.25195pt}{0.0pt}\pgfsys@curveto{4.25195pt}{2.34831pt}{2.34831pt}{4.25195pt}{0.0pt}{4.25195pt}\pgfsys@curveto{-2.34831pt}{4.25195pt}{-4.25195pt}{2.34831pt}{-4.25195pt}{0.0pt}\pgfsys@curveto{-4.25195pt}{-2.34831pt}{-2.34831pt}{-4.25195pt}{0.0pt}{-4.25195pt}\pgfsys@curveto{2.34831pt}{-4.25195pt}{4.25195pt}{-2.34831pt}{4.25195pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle+\underbrace{\left(\frac{1}{\tilde{P}[A]}-\frac{1}{P_{0}[A]}\right)\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[h(Z)]+\left(\frac{1}{\tilde{P}[A]}-\frac{1}{\mathbb{P}_{n}[A]}\right)\sqrt{n}\mathbb{P}_{n}[h(Z)].}_{\leavevmode\hbox
to8.9pt{\vbox to8.9pt{\pgfpicture\makeatletter\hbox{\hskip
4.45195pt\lower-4.45195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.25195pt}{0.0pt}\pgfsys@curveto{4.25195pt}{2.34831pt}{2.34831pt}{4.25195pt}{0.0pt}{4.25195pt}\pgfsys@curveto{-2.34831pt}{4.25195pt}{-4.25195pt}{2.34831pt}{-4.25195pt}{0.0pt}\pgfsys@curveto{-4.25195pt}{-2.34831pt}{-2.34831pt}{-4.25195pt}{0.0pt}{-4.25195pt}\pgfsys@curveto{2.34831pt}{-4.25195pt}{4.25195pt}{-2.34831pt}{4.25195pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$

We will show that term 1 converges conditionally to the requisite
$\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}})$ distribution, while term 2
converges conditionally to zero. Combining these two results with Lemma 10
completes the proof.

As before, it is sufficient to prove the result conditioned on $P\in H_{n}$.
For term 1, we have

$\displaystyle d_{BL}\left(\mathcal{L}\left(\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\mid P\in
H_{n},Z^{(n)}\right),\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}})\right)$
$\displaystyle\leq
d_{BL}\left(\mathcal{L}\left(\frac{\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[h(Z)]}{P_{0}[A]}\mid
P\in
H_{n},Z^{(n)}\right),\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}})\right)$
$\displaystyle+\sup_{P\in
H_{n}}\left|\frac{\mathbb{G}_{n}[h(Z)-h_{0}(Z)]}{\mathbb{P}_{n}[A]}\right|+\sup_{P\in
H_{n}}\left|\frac{\sqrt{n}P_{0}[h(Z)]}{\mathbb{P}_{n}[A]}\right|.$

Provided that

$\sup_{P\in H_{n}}\|h-h_{0}\|_{P_{0}}\rightarrow 0,$

the first two terms on the right can be handled in the same way as Theorem 2.
Namely, the first term uses Theorem 3.6.13 of van der Vaart and Wellner
[1996], and the second term uses Corollary 2.3.12 of van der Vaart and Wellner
[1996]. To establish the uniform $L_{2}$-convergence of $h$, consider

$\displaystyle h(Z)-h_{0}(Z)=$
$\displaystyle\left[\frac{A-\pi(X)}{1-\pi(X)}\right]\\{Y-\mu^{(0)}(X)\\}-\left[\frac{A-\pi_{0}(X)}{1-\pi_{0}(X)}\right]\\{Y-\mu_{0}^{(0)}(X)\\}$
$\displaystyle=$
$\displaystyle\left(1-\frac{1-A}{1-\pi(X)}\right)\\{\mu_{0}^{(0)}(X)-\mu^{(0)}(X)\\}+(1-A)\left[\frac{1}{1-\pi_{0}(X)}-\frac{1}{1-\pi(X)}\right]\\{Y-\mu_{0}^{(0)}(X)\\}$
$\displaystyle=$
$\displaystyle\left(1-\frac{1-A}{1-\pi(X)}\right)\\{\mu_{0}^{(0)}(X)-\mu^{(0)}(X)\\}+(1-A)\left[\frac{\pi_{0}(X)-\pi(X)}{(1-\pi_{0}(X))(1-\pi(X))}\right]\\{Y-\mu_{0}^{(0)}(X)\\}$

so

$\|h-h_{0}\|_{P_{0}}\leq\left(1+\frac{1}{\delta}\right)\|\mu_{0}^{(0)}-\mu^{(0)}\|_{P_{0}}+\left(\frac{C}{\delta^{2}}\right)\|\pi-\pi_{0}\|_{P_{0}},$

which establishes the uniform $L_{2}$-convergence of $h$ under our
assumptions.

For the final term in our upper bound for 1, we have

$\displaystyle P_{0}[h(Z)]=$ $\displaystyle
P_{0}\left[\left(\frac{A-\pi(X)}{1-\pi(X)}\right)\\{Y-\mu^{(0)}(X)\\}-\pi_{0}(X)\\{\mu_{0}^{(1)}(X)-\mu_{0}^{(0)}(X)\\}\right]$
$\displaystyle=$ $\displaystyle
P_{0}\left[\frac{\pi_{0}(X)\mu_{0}^{(1)}(X)-\pi_{0}(X)\mu^{(0)}(X)-\pi(X)[\pi_{0}(X)\mu_{0}^{(1)}(X)+(1-\pi_{0}(X))\mu_{0}^{(0)}(X)]+\pi(X)\mu^{(0)}(X)}{1-\pi(X)}\right]$
$\displaystyle-
P_{0}\left[\pi_{0}(X)\\{\mu_{0}^{(1)}(X)-\mu_{0}^{(0)}(X)\\}\right]$
$\displaystyle=$ $\displaystyle
P_{0}\left[\frac{-\pi_{0}(X)\mu^{(0)}(X)-\pi(X)(1-\pi_{0}(X))\mu_{0}^{(0)}(X)+\pi(X)\mu^{(0)}(X)+\pi_{0}(X)\mu_{0}^{(0)}(X)(1-\pi(X))}{1-\pi(X)}\right]$
$\displaystyle=$ $\displaystyle
P_{0}\left[\frac{-\pi_{0}(X)\mu^{(0)}(X)-\pi(X)\mu_{0}^{(0)}(X)+\pi(X)\mu^{(0)}(X)+\pi_{0}(X)\mu_{0}^{(0)}(X))}{1-\pi(X)}\right]$
$\displaystyle=$ $\displaystyle
P_{0}\left[\frac{(\pi(X)-\pi_{0}(X))(\mu^{(0)}(X)-\mu_{0}^{(0)}(X))}{1-\pi(X)}\right]$
$\displaystyle\leq$
$\displaystyle\frac{1}{\delta}\|\pi-\pi_{0}\|_{P_{0}}\|\mu_{0}^{(0)}-\mu^{(0)}\|_{P_{0}}$

where the inequality on the final line is due to Cauchy-Schwarz and the
assumed bounding on $\pi$. By assumption, the product of the errors converge
uniformly faster than $1/\sqrt{n}$.

Now we proceed to study term 2. First we note that $\tilde{P}[A]$ converges
conditionally in distribution to $P_{0}[A]$ almost surely under the infinite
product probability measure $P_{0}^{(\infty)}$. For an outcome in the infinite
product sample space such that $\tilde{P}[A]\rightsquigarrow P_{0}[A]$, we can
apply Slutsky’s lemma to deduce that

$\left(\frac{P_{0}[A]}{\tilde{P}[A]}-1\right)\rightsquigarrow 0.$

Therefore, this also holds conditionally almost surely $[P_{0}^{(\infty)}]$.
We already know that

$d_{BL}\left(\mathcal{L}\left(\frac{\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[h(Z)]}{P_{0}[A]}\mid
P\in
H_{n},Z^{(n)}\right),\mathcal{N}(0,\|\dot{\chi}_{P_{0}}\|^{2}_{P_{0}})\right)\xrightarrow[]{P_{0}}0$

so by Lemma 10, we deduce that

$d_{BL}\left(\mathcal{L}\left(\left(\frac{P_{0}[A]}{\tilde{P}[A]}-1\right)\frac{\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[h(Z)]}{P_{0}[A]}\mid
P\in H_{n},Z^{(n)}\right),\delta_{\\{0\\}}\right)\xrightarrow[]{P_{0}}0.$

Finally, we have the second term in 2. Using similar arguments to the above,
we have that

$\left(\frac{1}{\tilde{P}[A]}-\frac{1}{\mathbb{P}_{n}[A]}\right)$

converges conditionally in distribution to zero almost surely
$[P_{0}^{(\infty)}]$. We also have

$\sqrt{n}\mathbb{P}_{n}[h(Z)]=\sqrt{n}(\mathbb{P}_{n}-P_{0})[h(Z)-h_{0}(Z)]+\sqrt{n}P_{0}[h(Z)]+\sqrt{n}\mathbb{P}_{n}[h_{0}(Z)].$

We have already established that the first two terms on the right converge
uniformly to zero over $P\in H_{n}$ (the first one is in probability). The
final term converges weakly to a normal distribution by the central limit
theorem. Thus, by Lemma 10, we have

$d_{BL}\left(\mathcal{L}\left(\left(\frac{1}{\tilde{P}[A]}-\frac{1}{\mathbb{P}_{n}[A]}\right)\sqrt{n}\mathbb{P}_{n}[h(Z)]\mid
P\in H_{n},Z^{(n)}\right),\delta_{\\{0\\}}\right)\xrightarrow[]{P_{0}}0.$

We use Lemma 10 once more to combine the results for the two terms in 2. ∎

###  E.7 Proof of Theorem 13

Define

$\displaystyle g(Z)$
$\displaystyle=\left(A-\frac{(1-A)\pi(X)}{1-\pi(X)}\right)[Y-\mu^{(A)}(X)]+[A-\pi(X)]\\{\mu^{(1)}(X)-\mu^{(0)}(X)-\chi(P_{0})\\}$
$\displaystyle g_{0}(Z)$
$\displaystyle=\left(A-\frac{(1-A)\pi_{0}(X)}{1-\pi_{0}(X)}\right)[Y-\mu_{0}^{(A)}(X)]+[A-\pi_{0}(X)]\\{\mu_{0}^{(1)}(X)-\mu_{0}^{(0)}(X)-\chi(P_{0})\\}.$

We begin by establishing the uniform $L_{2}$-convergence of $g$ on $H_{n}$:

$\sup_{P\in H_{n}}\|g-g_{0}\|_{P_{0}}\rightarrow 0.$

Using the definitions of $h$ and $h_{0}$ from the proof of Theorem 8 in
Section E.6, $g$ and $g_{0}$ can be rewritten as

$\displaystyle g(Z)$
$\displaystyle=h(Z)-\pi(X)\\{\mu^{(1)}(X)-\mu^{(0)}(X)-\chi(P_{0})\\}$
$\displaystyle g_{0}(Z)$
$\displaystyle=h_{0}(Z)-\pi_{0}(X)\\{\mu_{0}^{(1)}(X)-\mu_{0}^{(0)}(X)-\chi(P_{0})\\}.$

Under the assumptions, we already showed that $h$ converges uniformly in
$L_{2}$ to $h_{0}$ on $H_{n}$. Thus, it suffices to show that

$\sup_{P\in
H_{n}}\|\pi\\{\mu^{(1)}-\mu^{(0)}-\chi(P_{0})\\}-\pi_{0}\\{\mu_{0}^{(1)}-\mu_{0}^{(0)}-\chi(P_{0})\\}\|_{P_{0}}\rightarrow
0.$

The expression inside the norm can be decomposed into

$\pi\\{\mu^{(1)}-\mu^{(0)}-(\mu_{0}^{(1)}-\mu_{0}^{(0)})\\}+(\pi-\pi_{0})\\{\mu_{0}^{(1)}-Y+Y-\mu_{0}^{(0)}-\chi(P_{0})\\}$

and this has $L_{2}$-norm upper-bounded by

$\|\mu^{(1)}-\mu_{0}^{(1)}\|_{P_{0}}+\|\mu^{(0)}-\mu_{0}^{(0)}\|_{P_{0}}+(2C+|\chi(P_{0})|)\|\pi-\pi_{0}\|_{P_{0}},$

which converges uniformly to zero on $H_{0}$ by Assumptions 8(a) and (c).

To reduce verbosity and repetition, we will say hereafter that an expression
is “controlled” if it converges conditionally to zero in probability on
$H_{n}$. The ACTT one-step corrected parameter can be decomposed as follows:

$\displaystyle\tilde{\theta}$
$\displaystyle=\theta(P)+\tilde{P}\left(\frac{A}{\tilde{P}[A]}-\frac{(1-A)\pi(X)}{\tilde{P}[A][1-\pi(X)]}\right)[Y-\mu^{(A)}(X)]+\tilde{P}\left[\left(\frac{A-\pi(X)}{\tilde{P}[A]}\right)\\{\mu^{(1)}(X)-\mu^{(0)}(X)-\theta(P)\\}\right]$
$\displaystyle=\frac{\tilde{P}[g(Z)]}{\tilde{P}[A]}+\chi(P_{0})+\frac{\tilde{P}[\pi(X)]\\{\theta(P)-\chi(P_{0})\\}}{\tilde{P}[A]}$

We can further decompose $\tilde{P}[g(Z)]$ into

$\displaystyle\tilde{P}[g(Z)]$
$\displaystyle=[\tilde{P}-\mathbb{P}_{n}][g(Z)]+\mathbb{P}_{n}[h(Z)]-\mathbb{P}_{n}[\pi(X)\\{\mu^{(1)}(X)-\mu^{(0)}(X)-\chi(P_{0})\\}]$
$\displaystyle=[\tilde{P}-\mathbb{P}_{n}][g(Z)]+\mathbb{P}_{n}[h(Z)]-\mathbb{P}_{n}[\pi(X)]\\{\theta(P)-\chi(P_{0})\\},$

where the second equality follows from the definition of $\theta(P)$, which
satisfies

$\mathbb{P}_{n}[\pi(X)\\{\mu^{(1)}(X)-\mu^{(0)}(X)-\theta(P)\\}]=0.$

Then

$\displaystyle\sqrt{n}(\tilde{\theta}-\hat{\chi}_{n})$
$\displaystyle=\underbrace{\frac{\sqrt{n}[\tilde{P}-\mathbb{P}_{n}][g(Z)]}{\tilde{P}[A]}}_{\leavevmode\hbox
to8.9pt{\vbox to8.9pt{\pgfpicture\makeatletter\hbox{\hskip
4.45195pt\lower-4.45195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.25195pt}{0.0pt}\pgfsys@curveto{4.25195pt}{2.34831pt}{2.34831pt}{4.25195pt}{0.0pt}{4.25195pt}\pgfsys@curveto{-2.34831pt}{4.25195pt}{-4.25195pt}{2.34831pt}{-4.25195pt}{0.0pt}\pgfsys@curveto{-4.25195pt}{-2.34831pt}{-2.34831pt}{-4.25195pt}{0.0pt}{-4.25195pt}\pgfsys@curveto{2.34831pt}{-4.25195pt}{4.25195pt}{-2.34831pt}{4.25195pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\underbrace{\frac{\sqrt{n}\mathbb{P}_{n}[h(Z)]}{\tilde{P}[A]}-\frac{\sqrt{n}\mathbb{P}_{n}[h_{0}(Z)]}{P_{0}[A]}}_{\leavevmode\hbox
to8.9pt{\vbox to8.9pt{\pgfpicture\makeatletter\hbox{\hskip
4.45195pt\lower-4.45195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.25195pt}{0.0pt}\pgfsys@curveto{4.25195pt}{2.34831pt}{2.34831pt}{4.25195pt}{0.0pt}{4.25195pt}\pgfsys@curveto{-2.34831pt}{4.25195pt}{-4.25195pt}{2.34831pt}{-4.25195pt}{0.0pt}\pgfsys@curveto{-4.25195pt}{-2.34831pt}{-2.34831pt}{-4.25195pt}{0.0pt}{-4.25195pt}\pgfsys@curveto{2.34831pt}{-4.25195pt}{4.25195pt}{-2.34831pt}{4.25195pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}-\underbrace{\frac{\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\pi(X)]\\{\theta(P)-\chi(P_{0})\\}}{\tilde{P}[A]}}_{\leavevmode\hbox
to8.9pt{\vbox to8.9pt{\pgfpicture\makeatletter\hbox{\hskip
4.45195pt\lower-4.45195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.25195pt}{0.0pt}\pgfsys@curveto{4.25195pt}{2.34831pt}{2.34831pt}{4.25195pt}{0.0pt}{4.25195pt}\pgfsys@curveto{-2.34831pt}{4.25195pt}{-4.25195pt}{2.34831pt}{-4.25195pt}{0.0pt}\pgfsys@curveto{-4.25195pt}{-2.34831pt}{-2.34831pt}{-4.25195pt}{0.0pt}{-4.25195pt}\pgfsys@curveto{2.34831pt}{-4.25195pt}{4.25195pt}{-2.34831pt}{4.25195pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
We will show that term 1 converges conditionally in distribution to
$\mathcal{N}(0,\|\dot{\chi}^{Y}_{P_{0}}+\dot{\chi}^{A}_{P_{0}}\|^{2}_{P_{0}})$
as required, while the other two terms are controlled.

Term 1 can be decomposed into

$\frac{\sqrt{n}[\tilde{P}-\mathbb{P}_{n}][g(Z)]}{\tilde{P}[A]}=\frac{\sqrt{n}[\tilde{P}-\mathbb{P}_{n}][g(Z)]}{P_{0}[A]}+\left(\frac{P_{0}[A]}{\tilde{P}[A]}-1\right)\frac{\sqrt{n}[\tilde{P}-\mathbb{P}_{n}][g(Z)]}{P_{0}[A]}.$

For the first term on the right, we already showed that $g$ converges
uniformly in $L_{2}$ to $g_{0}$ on $H_{n}$, and moreover,

$\frac{g_{0}}{P_{0}[A]}=\dot{\chi}^{Y}_{P_{0}}+\dot{\chi}^{A}_{P_{0}}.$

As argued in Theorem 1, the Donsker class assumption establishes the necessary
Bayesian bootstrap Bernstein-von Mises theorem, which implies the convergence
to
$\mathcal{N}(0,\|\dot{\chi}^{Y}_{P_{0}}+\dot{\chi}^{A}_{P_{0}}\|^{2}_{P_{0}})$.
For the second term, we already showed in the proof of Theorem 8 that the
factor $(P_{0}[A]/\tilde{P}[A]-1)$ converges conditionally in distribution to
zero almost surely. Thus, by Lemma 10, this second term is controlled.

For term 2, we have

$\displaystyle\frac{\sqrt{n}\mathbb{P}_{n}[h(Z)]}{\tilde{P}[A]}-\frac{\sqrt{n}\mathbb{P}_{n}[h_{0}(Z)]}{P_{0}[A]}$
$\displaystyle=\underbrace{\frac{\sqrt{n}(\mathbb{P}_{n}-P_{0})[h(Z)-h_{0}(Z)]}{P_{0}[A]}}_{\leavevmode\hbox
to10.32pt{\vbox to10.32pt{\pgfpicture\makeatletter\hbox{\hskip
5.1624pt\lower-5.1624pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.9624pt}{0.0pt}\pgfsys@curveto{4.9624pt}{2.74069pt}{2.74069pt}{4.9624pt}{0.0pt}{4.9624pt}\pgfsys@curveto{-2.74069pt}{4.9624pt}{-4.9624pt}{2.74069pt}{-4.9624pt}{0.0pt}\pgfsys@curveto{-4.9624pt}{-2.74069pt}{-2.74069pt}{-4.9624pt}{0.0pt}{-4.9624pt}\pgfsys@curveto{2.74069pt}{-4.9624pt}{4.9624pt}{-2.74069pt}{4.9624pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.625pt}{-2.39166pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{A}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\underbrace{\frac{\sqrt{n}P_{0}[h(Z)]}{P_{0}[A]}}_{\leavevmode\hbox
to10.08pt{\vbox to10.08pt{\pgfpicture\makeatletter\hbox{\hskip
5.04057pt\lower-5.04057pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.84058pt}{0.0pt}\pgfsys@curveto{4.84058pt}{2.6734pt}{2.6734pt}{4.84058pt}{0.0pt}{4.84058pt}\pgfsys@curveto{-2.6734pt}{4.84058pt}{-4.84058pt}{2.6734pt}{-4.84058pt}{0.0pt}\pgfsys@curveto{-4.84058pt}{-2.6734pt}{-2.6734pt}{-4.84058pt}{0.0pt}{-4.84058pt}\pgfsys@curveto{2.6734pt}{-4.84058pt}{4.84058pt}{-2.6734pt}{4.84058pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.47917pt}{-2.39166pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{B}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle+\underbrace{\left(\frac{P_{0}[A]}{\tilde{P}[A]}-1\right)\left(\frac{\sqrt{n}\mathbb{P}_{n}[h_{0}(Z)]}{P_{0}[A]}+\frac{\sqrt{n}(\mathbb{P}_{n}-P_{0})[h(Z)-h_{0}(Z)]}{P_{0}[A]}+\frac{\sqrt{n}P_{0}[h(Z)]}{P_{0}[A]}\right).}_{\leavevmode\hbox
to10.16pt{\vbox to10.16pt{\pgfpicture\makeatletter\hbox{\hskip
5.08159pt\lower-5.08159pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.88159pt}{0.0pt}\pgfsys@curveto{4.88159pt}{2.69606pt}{2.69606pt}{4.88159pt}{0.0pt}{4.88159pt}\pgfsys@curveto{-2.69606pt}{4.88159pt}{-4.88159pt}{2.69606pt}{-4.88159pt}{0.0pt}\pgfsys@curveto{-4.88159pt}{-2.69606pt}{-2.69606pt}{-4.88159pt}{0.0pt}{-4.88159pt}\pgfsys@curveto{2.69606pt}{-4.88159pt}{4.88159pt}{-2.69606pt}{4.88159pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.52777pt}{-2.39166pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{C}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$

Terms A and B were already shown to be controlled in the proof of Theorem 8.
As before, the factor $(P_{0}[A]/\tilde{P}[A]-1)$ converges conditionally in
distribution to zero almost surely. By the central limit theorem,
$\sqrt{n}\mathbb{P}_{n}[h_{0}(Z)]$ converges weakly to a normal distribution
with finite variance. The second and third terms in the second bracket of term
C are terms A and B respectively. We deduce that term C is controlled as well.

Finally, for term 3, we have

$\leavevmode\hbox to11.37pt{\vbox
to11.37pt{\pgfpicture\makeatletter\hbox{\hskip 5.68657pt\lower-5.68657pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.48657pt}{0.0pt}\pgfsys@curveto{5.48657pt}{3.03018pt}{3.03018pt}{5.48657pt}{0.0pt}{5.48657pt}\pgfsys@curveto{-3.03018pt}{5.48657pt}{-5.48657pt}{3.03018pt}{-5.48657pt}{0.0pt}\pgfsys@curveto{-5.48657pt}{-3.03018pt}{-3.03018pt}{-5.48657pt}{0.0pt}{-5.48657pt}\pgfsys@curveto{3.03018pt}{-5.48657pt}{5.48657pt}{-3.03018pt}{5.48657pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\pi(X)]\\{\theta(P)-\chi(P_{0})\\}}{P_{0}[A]}+\left(\frac{P_{0}[A]}{\tilde{P}[A]}-1\right)\frac{\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\pi(X)]\\{\theta(P)-\chi(P_{0})\\}}{P_{0}[A]}.$

Therefore, it suffices to show that
$\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\pi(X)]\\{\theta(P)-\chi(P_{0})\\}$ is
controlled. Since $\pi$ is eventually contained in a fixed $P_{0}$-Donsker
class and converges uniformly in $L_{2}$ to $\pi_{0}$, we have that
$\sqrt{n}(\tilde{P}-\mathbb{P}_{n})[\pi(X)]$ converges conditionally in
distribution to $\mathcal{N}(0,P_{0}[(\pi_{0}-P_{0}[\pi_{0}])^{2}])$ in
probability.

It remains to study the convergence of $\theta(P)$ to $\chi(P_{0})$:

$\displaystyle\theta(P)-\chi(P_{0})$
$\displaystyle=\frac{\mathbb{Q}_{n}[\pi\\{\mu^{(1)}-\mu^{(0)}\\}]}{\mathbb{Q}_{n}[\pi]}-\frac{P_{0}[\pi_{0}\\{\mu_{0}^{(1)}-\mu_{0}^{(0)}\\}]}{P_{0}[\pi_{0}]}$
$\displaystyle=\underbrace{\left(\frac{P_{0}[\pi_{0}]}{\mathbb{Q}_{n}[\pi]}-1\right)\frac{\mathbb{Q}_{n}[\pi\\{\mu^{(1)}-\mu^{(0)}\\}]}{P_{0}[\pi_{0}]}}_{\leavevmode\hbox
to8.55pt{\vbox to8.55pt{\pgfpicture\makeatletter\hbox{\hskip
4.27617pt\lower-4.27617pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.07617pt}{0.0pt}\pgfsys@curveto{4.07617pt}{2.25124pt}{2.25124pt}{4.07617pt}{0.0pt}{4.07617pt}\pgfsys@curveto{-2.25124pt}{4.07617pt}{-4.07617pt}{2.25124pt}{-4.07617pt}{0.0pt}\pgfsys@curveto{-4.07617pt}{-2.25124pt}{-2.25124pt}{-4.07617pt}{0.0pt}{-4.07617pt}\pgfsys@curveto{2.25124pt}{-4.07617pt}{4.07617pt}{-2.25124pt}{4.07617pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.26389pt}{-2.39166pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{I}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\underbrace{\frac{\mathbb{Q}_{n}[\pi\\{\mu^{(1)}-\mu^{(0)}\\}]-P_{0}[\pi_{0}\\{\mu_{0}^{(1)}-\mu_{0}^{(0)}\\}]}{P_{0}[\pi_{0}]}}_{\leavevmode\hbox
to10.32pt{\vbox to10.32pt{\pgfpicture\makeatletter\hbox{\hskip
5.1624pt\lower-5.1624pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.9624pt}{0.0pt}\pgfsys@curveto{4.9624pt}{2.74069pt}{2.74069pt}{4.9624pt}{0.0pt}{4.9624pt}\pgfsys@curveto{-2.74069pt}{4.9624pt}{-4.9624pt}{2.74069pt}{-4.9624pt}{0.0pt}\pgfsys@curveto{-4.9624pt}{-2.74069pt}{-2.74069pt}{-4.9624pt}{0.0pt}{-4.9624pt}\pgfsys@curveto{2.74069pt}{-4.9624pt}{4.9624pt}{-2.74069pt}{4.9624pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.625pt}{-2.39166pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{II}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
First consider term II. The numerator can be decomposed into

$\displaystyle P_{0}[\pi_{0}]\leavevmode\hbox to13.34pt{\vbox
to13.34pt{\pgfpicture\makeatletter\hbox{\hskip 6.66801pt\lower-6.66801pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.46802pt}{0.0pt}\pgfsys@curveto{6.46802pt}{3.57222pt}{3.57222pt}{6.46802pt}{0.0pt}{6.46802pt}\pgfsys@curveto{-3.57222pt}{6.46802pt}{-6.46802pt}{3.57222pt}{-6.46802pt}{0.0pt}\pgfsys@curveto{-6.46802pt}{-3.57222pt}{-3.57222pt}{-6.46802pt}{0.0pt}{-6.46802pt}\pgfsys@curveto{3.57222pt}{-6.46802pt}{6.46802pt}{-3.57222pt}{6.46802pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{II}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=(\mathbb{Q}_{n}-P_{0})[\pi\\{\mu^{(1)}-\mu^{(0)}\\}]+P_{0}[\pi\\{\mu^{(1)}-\mu^{(0)}\\}-\pi_{0}\\{\mu_{0}^{(1)}-\mu_{0}^{(0)}\\}].$

Define the classes $\mathcal{H}_{n,1}=\\{\pi:P\in H_{n}\\}$,
$\mathcal{H}_{n,2}=\\{\mu^{(1)}-y:P\in H_{n}\\}$ and
$\mathcal{H}_{n,3}=\\{\mu^{(0)}-y:P\in H_{n}\\}$. These have envelope
functions $1$, $C$ and $C$ respectively. The first and third classes are each
eventually contained in fixed $P_{0}$-Glivenko-Cantelli classes by assumption.
We apply Lemma 11 of Ray and van der Vaart [2020] with the continuous function
$\phi(h_{1},h_{2},h_{3})=h_{1}(h_{2}-h_{3})$ to deduce that

$\sup_{P\in
H_{n}}|(\mathbb{Q}_{n}-P_{0})[\pi\\{\mu^{(1)}-\mu^{(0)}\\}]|\xrightarrow[]{P_{0}^{*}}0.$

For the remaining term in the numerator of II, consider the decomposition

$\displaystyle|P_{0}[\pi\\{\mu^{(1)}-\mu^{(0)}\\}-\pi_{0}\\{\mu_{0}^{(1)}-\mu_{0}^{(0)}\\}]|$
$\displaystyle=|P_{0}[\pi\\{\mu^{(1)}-\mu^{(0)}-\mu_{0}^{(1)}+\mu_{0}^{(0)}\\}]-P_{0}[(\pi_{0}-\pi)\\{\mu_{0}^{(1)}-Y+Y-\mu_{0}^{(0)}\\}]|$
$\displaystyle\leq
P_{0}|\mu^{(1)}-\mu_{0}^{(1)}|+P_{0}|\mu^{0)}-\mu_{0}^{(0)}|+2CP_{0}|\pi-\pi_{0}|$
$\displaystyle\leq\|\mu^{(1)}-\mu_{0}^{(1)}\|_{P_{0}}+\|\mu^{0)}-\mu_{0}^{(0)}\|_{P_{0}}+2C\|\pi-\pi_{0}\|_{P_{0}}.$

The final inequality follows from Cauchy-Schwarz. Thus, term II is controlled.

For term I, the first bracket is upper-bounded by

$\displaystyle\left|\frac{P_{0}[\pi_{0}]}{\mathbb{Q}_{n}[\pi]}-1\right|$
$\displaystyle=\left|\frac{P_{0}[\pi_{0}]-\mathbb{Q}_{n}[\pi]}{\mathbb{Q}_{n}[\pi]}\right|$
$\displaystyle\leq\frac{1}{\delta}\left|P_{0}[\pi_{0}]-\mathbb{Q}_{n}[\pi]\right|$
$\displaystyle\leq\frac{1}{\delta}P_{0}\left|\pi_{0}-\pi\right|+\frac{1}{\delta}\left|(\mathbb{Q}_{n}-P_{0})[\pi]\right|$
$\displaystyle\leq\frac{1}{\delta}\|\pi_{0}-\pi\|_{P_{0}}+\frac{1}{\delta}\left|(\mathbb{Q}_{n}-P_{0})[\pi]\right|.$

The upper bound is controlled since $\pi$ converges uniformly to $\pi_{0}$ in
$L_{2}$ on $H_{n}$, and $\pi$ is eventually contained in a $P_{0}$-Glivenko-
Cantelli class. We can write term I as

$\leavevmode\hbox to10.87pt{\vbox
to10.87pt{\pgfpicture\makeatletter\hbox{\hskip 5.43437pt\lower-5.43437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.23438pt}{0.0pt}\pgfsys@curveto{5.23438pt}{2.8909pt}{2.8909pt}{5.23438pt}{0.0pt}{5.23438pt}\pgfsys@curveto{-2.8909pt}{5.23438pt}{-5.23438pt}{2.8909pt}{-5.23438pt}{0.0pt}\pgfsys@curveto{-5.23438pt}{-2.8909pt}{-2.8909pt}{-5.23438pt}{0.0pt}{-5.23438pt}\pgfsys@curveto{2.8909pt}{-5.23438pt}{5.23438pt}{-2.8909pt}{5.23438pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.80556pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{I}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\left(\frac{P_{0}[\pi_{0}]}{\mathbb{Q}_{n}[\pi]}-1\right)\left(\leavevmode\hbox
to13.34pt{\vbox to13.34pt{\pgfpicture\makeatletter\hbox{\hskip
6.66801pt\lower-6.66801pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.46802pt}{0.0pt}\pgfsys@curveto{6.46802pt}{3.57222pt}{3.57222pt}{6.46802pt}{0.0pt}{6.46802pt}\pgfsys@curveto{-3.57222pt}{6.46802pt}{-6.46802pt}{3.57222pt}{-6.46802pt}{0.0pt}\pgfsys@curveto{-6.46802pt}{-3.57222pt}{-3.57222pt}{-6.46802pt}{0.0pt}{-6.46802pt}\pgfsys@curveto{3.57222pt}{-6.46802pt}{6.46802pt}{-3.57222pt}{6.46802pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{II}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\frac{P_{0}[\pi_{0}\\{\mu_{0}^{(1)}-\mu_{0}^{(0)}\\}]}{P_{0}[\pi_{0}]}\right).$

Since we have already showed that II is controlled, we deduce that term I is
also controlled. This completes the proof. ∎