[proofplan]
We compare the empirically centered multiplier process with the multiplier process centered by the true mean $Pf$. The standard multiplier [central limit theorem](/theorems/521) for Donsker classes gives the conditional [weak convergence](/page/Weak%20Convergence) of the $Pf$-centered process. The difference between the two processes factors into the product of the scalar multiplier average and the uniform empirical deviation $\sup_{f\in\mathcal F}|P_nf-Pf|$. The scalar factor has bounded second moment, while the Donsker property gives $\sqrt n\sup_{f\in\mathcal F}|P_nf-Pf|=O_{\mathbb P_X}(1)$, so the correction is conditionally negligible.
[/proofplan]
[step:Introduce the true-mean centered multiplier process]
For each $n\in\mathbb N$, define
\begin{align*}
H_n^\xi:\Omega\to\ell^\infty(\mathcal F)
\end{align*}
by
\begin{align*}
H_n^\xi(f):=\frac{1}{\sqrt n}\sum_{i=1}^n \xi_i\bigl(f(X_i)-Pf\bigr)
\end{align*}
for $f\in\mathcal F$. By the standard multiplier [central limit theorem](/theorems/1848) for Donsker classes under the multiplier tail condition
\begin{align*}
\int_0^\infty \sqrt{\mathbb P_\xi(|\xi_1|>t)}\,d\mathcal L^1(t)<\infty,
\end{align*}
(citing a result not yet in the wiki: Multiplier central limit theorem for Donsker classes), applied to the $P$-Donsker class $\mathcal F$ with square-integrable envelope $F$, we have
\begin{align*}
H_n^\xi\rightsquigarrow G_P
\end{align*}
conditionally in probability in $\ell^\infty(\mathcal F)$, conditionally on $X_1,\dots,X_n$.
[/step]
[step:Factor the empirical-centering correction]
Define the scalar multiplier average
\begin{align*}
S_n^\xi:\Omega_\xi\to\mathbb R
\end{align*}
by
\begin{align*}
S_n^\xi:=\frac{1}{\sqrt n}\sum_{i=1}^n \xi_i.
\end{align*}
For every $f\in\mathcal F$,
\begin{align*}
G_n^\xi(f)=H_n^\xi(f)-S_n^\xi(P_nf-Pf).
\end{align*}
Indeed,
\begin{align*}
f(X_i)-P_nf=\bigl(f(X_i)-Pf\bigr)-\bigl(P_nf-Pf\bigr)
\end{align*}
for each $1\le i\le n$, and summing after multiplying by $\xi_i/\sqrt n$ gives the displayed identity.
Consequently, if
\begin{align*}
R_n^\xi:\Omega\to\ell^\infty(\mathcal F)
\end{align*}
is defined by
\begin{align*}
R_n^\xi(f):=G_n^\xi(f)-H_n^\xi(f),
\end{align*}
then
\begin{align*}
\|R_n^\xi\|_{\ell^\infty(\mathcal F)}
\le |S_n^\xi|\sup_{f\in\mathcal F}|P_nf-Pf|.
\end{align*}
[guided]
The only difference between $G_n^\xi$ and $H_n^\xi$ is the point around which the functions are centered. The process $H_n^\xi$ subtracts the deterministic mean $Pf$, while $G_n^\xi$ subtracts the empirical mean $P_nf$. We isolate this difference because the multiplier central limit theorem is naturally stated for the deterministic centering.
Define
\begin{align*}
S_n^\xi:\Omega_\xi\to\mathbb R
\end{align*}
by
\begin{align*}
S_n^\xi:=\frac{1}{\sqrt n}\sum_{i=1}^n \xi_i.
\end{align*}
For a fixed $f\in\mathcal F$, the algebraic identity
\begin{align*}
f(X_i)-P_nf=\bigl(f(X_i)-Pf\bigr)-\bigl(P_nf-Pf\bigr)
\end{align*}
holds for every $1\le i\le n$. Multiplying by $\xi_i/\sqrt n$ and summing over $i$ gives
\begin{align*}
\frac{1}{\sqrt n}\sum_{i=1}^n \xi_i\bigl(f(X_i)-P_nf\bigr)
=
\frac{1}{\sqrt n}\sum_{i=1}^n \xi_i\bigl(f(X_i)-Pf\bigr)
-
\left(\frac{1}{\sqrt n}\sum_{i=1}^n \xi_i\right)(P_nf-Pf).
\end{align*}
By the definitions of $G_n^\xi$, $H_n^\xi$, and $S_n^\xi$, this is exactly
\begin{align*}
G_n^\xi(f)=H_n^\xi(f)-S_n^\xi(P_nf-Pf).
\end{align*}
Now define
\begin{align*}
R_n^\xi:\Omega\to\ell^\infty(\mathcal F)
\end{align*}
by
\begin{align*}
R_n^\xi(f):=G_n^\xi(f)-H_n^\xi(f).
\end{align*}
Taking the supremum over $f\in\mathcal F$ and using the elementary inequality $|ab|\le |a||b|$ in $\mathbb R$, we obtain
\begin{align*}
\|R_n^\xi\|_{\ell^\infty(\mathcal F)}
=
\sup_{f\in\mathcal F}|S_n^\xi(P_nf-Pf)|
\le
|S_n^\xi|\sup_{f\in\mathcal F}|P_nf-Pf|.
\end{align*}
This factorization is the useful form: the first factor depends only on the multipliers, and the second factor is the ordinary empirical-process size.
[/guided]
[/step]
[step:Show the empirical-centering correction is conditionally negligible]
Define the empirical-process norm
\begin{align*}
\Delta_n:\Omega_X\to[0,\infty]
\end{align*}
by
\begin{align*}
\Delta_n:=\sup_{f\in\mathcal F}|P_nf-Pf|.
\end{align*}
Since $\mathcal F$ is $P$-Donsker, the empirical process
\begin{align*}
\alpha_n:\Omega_X\to\ell^\infty(\mathcal F)
\end{align*}
defined by
\begin{align*}
\alpha_n(f):=\sqrt n(P_nf-Pf)
\end{align*}
converges weakly in $\ell^\infty(\mathcal F)$ to $G_P$. Hence
\begin{align*}
\|\alpha_n\|_{\ell^\infty(\mathcal F)}=O_{\mathbb P_X}(1),
\end{align*}
and therefore
\begin{align*}
\Delta_n=O_{\mathbb P_X}(n^{-1/2}).
\end{align*}
For every $n\in\mathbb N$, independence and the moment assumptions on $\xi_1$ give
\begin{align*}
\mathbb E_\xi[S_n^\xi]=0
\end{align*}
and
\begin{align*}
\mathbb E_\xi[(S_n^\xi)^2]=1.
\end{align*}
Fix $\varepsilon>0$. Conditional on $X_1,\dots,X_n$, the quantity $\Delta_n$ is deterministic and $S_n^\xi$ is independent of $X_1,\dots,X_n$. By [Chebyshev's inequality](/theorems/1126) with the real-valued [random variable](/page/Random%20Variable) $S_n^\xi$,
\begin{align*}
\mathbb P_\xi\left(|S_n^\xi|\Delta_n>\varepsilon\mid X_1,\dots,X_n\right)
\le
\frac{\Delta_n^2}{\varepsilon^2}.
\end{align*}
Because $\Delta_n=O_{\mathbb P_X}(n^{-1/2})$, we have $\Delta_n^2\xrightarrow{\mathbb P_X}0$. Hence
\begin{align*}
\mathbb P_\xi\left(\|R_n^\xi\|_{\ell^\infty(\mathcal F)}>\varepsilon\mid X_1,\dots,X_n\right)
\xrightarrow{\mathbb P_X}0.
\end{align*}
Thus
\begin{align*}
R_n^\xi\xrightarrow{\mathbb P}0
\end{align*}
conditionally in probability in $\ell^\infty(\mathcal F)$.
[/step]
[step:Transfer the conditional limit from true centering to empirical centering]
Let $d_{\mathrm{BL}}$ denote the bounded-Lipschitz metric on probability measures on $\ell^\infty(\mathcal F)$, and let $\mathcal L_\xi(Z\mid X_1,\dots,X_n)$ denote the conditional law of an $\ell^\infty(\mathcal F)$-valued random element $Z$ given the sample. The conditional convergence of $H_n^\xi$ gives
\begin{align*}
d_{\mathrm{BL}}\left(\mathcal L_\xi(H_n^\xi\mid X_1,\dots,X_n),\mathcal L(G_P)\right)
\xrightarrow{\mathbb P_X}0.
\end{align*}
The conditional negligibility of $R_n^\xi$ implies, by the conditional Slutsky principle for convergence in $\ell^\infty(\mathcal F)$,
\begin{align*}
d_{\mathrm{BL}}\left(\mathcal L_\xi(H_n^\xi+R_n^\xi\mid X_1,\dots,X_n),\mathcal L(G_P)\right)
\xrightarrow{\mathbb P_X}0.
\end{align*}
Since $G_n^\xi=H_n^\xi+R_n^\xi$, this is precisely
\begin{align*}
G_n^\xi\rightsquigarrow G_P
\end{align*}
conditionally in probability in $\ell^\infty(\mathcal F)$. This proves the theorem.
[/step]
