\chapter{Likelihood ratios on a common sphere}\label{sec:likelihood}

This section compares two Gaussian laws conditioned on the same sphere, $\mu_v$ and $\mu_{\tilde v}$ with $\sum_iv_i=\sum_i\tilde v_i$.
We write $\tilde v_i=v_i(1+e_i)$, so that $I=\cI(v,\tilde v)=\frac12\sum_ie_i^2$ is the information of $\tilde v$ relative to $v$ (\cref{def:model-information}).
The main result, \cref{gp-pinned-lan-quantitative-third}, states that under $\mu_v$ the logarithm of the likelihood ratio $\dd\mu_{\tilde v}/\dd\mu_v$ is close in the Wasserstein distance $W_1$ to the Gaussian law $\cN(-I/2,I)$, with an error $C_B(\eta_v+\max_i|e_i|)$ that is uniform over $I\le B$, including $I=0$.
As a consequence, $\|\mu_{\tilde v}-\mu_v\|_{\TV}$ is close to $2\Phi(\sqrt I/2)-1$.
With $I=e^{-4u}$, this is the Gaussian profile $\Psi(u)$ of \cref{sec:model}, which is how $\Psi$ arises at high temperature.
In \cref{gp-pinned-lan}, we deduce the corresponding limit statements, including convergence of exponential moments and the case of diverging information.

The proof has four ingredients: an identity expressing total variation through the likelihood ratio (\cref{lem:lik-tv}); a bound on inverse moments of a positive quadratic form under $\mu_v$ (\cref{gp-pin-radial-inverse-fourth}); an exact integration by parts formula for quadratic statistics under $\mu_v$ (\cref{gp-pin-tangent-ibp-fourth}); and Stein's method, which turns this formula into a normal approximation for quadratic statistics (\cref{gp-scalar-stein-tests-fourth,gp-conditional-normal-third}).
The log likelihood ratio is a quadratic statistic plus a constant, and the information $I$ is its asymptotic variance.
We use \cref{gp-local-limit,gp-head-density,gp-growing-head,gp-real-tilt,gp-quadratic-lp,gp-quadratic-variance} from \cref{sec:conditioning}.
The results are used in the high-temperature profiles of \cref{sec:hot-fixed,sec:hot-meso,sec:infinite} and in the outer limit of \cref{sec:outer-hot}.

The Gaussian law $\cN(-I/2,I)$ and the profile $2\Phi(\sqrt I/2)-1$ are those of the Gaussian shift experiment in Le Cam's theory of local asymptotic normality \cite[Section~2, pp.~40--43]{LeCam1960}; see \cite[Chapters~6--7]{vanDerVaart1998}.
The results below concern the log likelihood ratio of two fixed laws, or of finitely many, and do not involve a parametrized family of laws.
Normal approximation by an integration by parts identity is Stein's method \cite{Stein1972}; see \cite[Sections~3.1--3.5]{NourdinPeccati2012} and \cite[Chapter~2]{ChenGoldsteinShao2011} for modern accounts.

\paragraph{Conventions.}
Throughout this section, all variance sequences are finite, indexed by $\{1,\dots,n\}$, and $\mu_v$ is the law of \cref{def:cond-finite} on the sphere $\mathcal S_{R^2}$, $R^2=\sum_iv_i$.
We keep the notation $S_v$, $\hat v_i$, $\eta_v$ of \eqref{gp-standardization} and write $Z_i=Y_i/\sqrt{v_i}$ under $\mu_v$.
Recall that $|Y|^2=R^2$ under $\mu_v$; dividing $|Y|^2-R^2=\sum_iv_i(Z_i^2-1)$ by $S_v$, we obtain the identity
\begin{equation}\label{eq:lik-constraint}
\sum_{i=1}^n\hat v_i\big(Z_i^2-1\big)=0\qquad\text{everywhere on }\mathcal S_{R^2}.
\end{equation}
For probability measures $\lambda_1,\lambda_2$ on $\R$ with finite first moments, $W_1(\lambda_1,\lambda_2)=\sup_h|\int h\dd\lambda_1-\int h\dd\lambda_2|$, the supremum being over $1$-Lipschitz functions $h$.
We write $\Law_\mu(X)$ for the law of $X$ under $\mu$, and $\cN(m,0)$ is the point mass at $m$.
For $w\in\R^n$, $\|w\|_2$ and $\|w\|_\infty$ are the Euclidean and maximum norms, and $\langle\cdot,\cdot\rangle$ is the Euclidean inner product.

\section{Total variation and likelihood ratios}\label{sec:lik-tv}

\begin{lemma}[Total variation through the likelihood ratio]\label{lem:lik-tv}
Let $P,Q$ be probability measures on a common space, with $Q$ absolutely continuous with respect to $P$, and let $\ell=\log(\dd Q/\dd P)\in[-\infty,\infty)$.
Then
\begin{equation}\label{gp-tv-bounded-test}
\|P-Q\|_{\TV}=\E_P\big(1-e^{\ell}\big)_+ .
\end{equation}
Let $I\ge0$ and let $X$ have law $\cN(-I/2,I)$.
Then
\begin{equation}\label{eq:lik-gaussian-profile}
\E\big(1-e^X\big)_+=2\Phi\big(\sqrt I/2\big)-1,\qquad \E e^{tX}=e^{(t^2-t)I/2}\quad(t\in\R).
\end{equation}
Further, the function $x\mapsto(1-e^x)_+$ is bounded by one and $1$-Lipschitz on $\R$, which gives $|\E_P(1-e^\ell)_+-\E(1-e^X)_+|\le W_1(\Law_P(\ell),\cN(-I/2,I))$ whenever $\ell$ is real-valued and integrable under $P$.
\end{lemma}

Note that $\cN(-I/2,I)$ is the law of $\log(\dd\cN(\sqrt I,1)/\dd\cN(0,1))$ under $\cN(0,1)$, so that \eqref{gp-tv-bounded-test} and \eqref{eq:lik-gaussian-profile} imply that $\|\cN(\sqrt I,1)-\cN(0,1)\|_{\TV}=2\Phi(\sqrt I/2)-1$.

\begin{proof}
Let $L=\dd Q/\dd P=e^\ell$.
Note that $\E_P(1-L)=0$, so that $\E_P(1-L)_+=\E_P(1-L)_-=\frac12\E_P|1-L|$, and $\frac12\E_P|1-L|=\frac12\int|\dd P-\dd Q|=\|P-Q\|_{\TV}$.
This proves \eqref{gp-tv-bounded-test}.
The first identity in \eqref{eq:lik-gaussian-profile} is immediate at $I=0$, when $X=0$; for its proof we may therefore assume $I>0$.
We write $X=-I/2+\sqrt I\,G$ and $x=\sqrt I/2$, so that $X<0$ exactly when $G<x$.
Completing the square, we have $e^{-I/2+\sqrt Ig}\phi(g)=\phi(g-\sqrt I)$, and
\[
\E\big(1-e^X\big)_+=\Phi(x)-\int_{-\infty}^x\phi\big(g-\sqrt I\big)\dd g=\Phi(x)-\Phi(x-\sqrt I)=2\Phi(x)-1 ,
\]
where we used $x-\sqrt I=-x$.
The second identity in \eqref{eq:lik-gaussian-profile} is the Gaussian moment generating function.
The function $(1-e^x)_+$ vanishes for $x\ge0$ and has derivative $-e^x\in[-1,0)$ for $x<0$, which implies the last statement.
\end{proof}

\section{Integration by parts under the conditioned law}\label{sec:lik-ibp}

\begin{lemma}[Inverse moments of a quadratic form]\label{gp-pin-radial-inverse-fourth}
There exist universal constants $\eta_{\mathrm{inv}}>0$ and $C<\infty$ such that every finite variance sequence $v$ with $\eta_v\le\eta_{\mathrm{inv}}$ satisfies
\begin{equation}\label{gp-pin-radial-inverse-bound-fourth}
\E_{\mu_v}V^{-p}\le C,\qquad p=1,2,\qquad\text{where}\quad V=\sum_{i=1}^n\hat v_i^2Z_i^2 .
\end{equation}
\end{lemma}

The constant does not depend on the dimension or on the smallest variance.
Note that $V>0$ everywhere on $\mathcal S_{R^2}$.

\begin{proof}
We fix $\eta_{\mathrm{inv}}$ such that $\eta_{\mathrm{inv}}^2\le1/48$ and $\eta_{\mathrm{inv}}$ is at most the constant $\eta_*$ of \cref{gp-growing-head}.
We add indices $1,2,\dots$ to a set $J$ until $q_J=\sum_{i\in J}\hat v_i^2$ first reaches $1/6$.
This happens, since $\sum_i\hat v_i^2=1/2$, and since the last index added contributes at most $\eta_v^2\le1/48$, we have $1/6\le q_J\le3/16$.
The fraction of the variance of $T_v$ carried by $J$ is $h_J=2q_J\le3/8\le1/2$, and \cref{gp-growing-head} implies that $r_{v,J}\le C$.
Using $V\ge\sum_{i\in J}\hat v_i^2Z_i^2$ and \cref{gp-head-density}, we obtain
\[
\E_{\mu_v}V^{-p}\le\int\Big(\sum_{i\in J}\hat v_i^2z_i^2\Big)^{-p}r_{v,J}(z)\,\gamma_J(\dd z)
\le C\,\E\Big(\sum_{i\in J}\hat v_i^2g_i^2\Big)^{-p}.
\]
Using $x^{-p}=\Gamma(p)^{-1}\int_0^\infty t^{p-1}e^{-tx}\dd t$, Tonelli's theorem, and $\E e^{-sg^2}=(1+2s)^{-1/2}$, we have
\[
\E\Big(\sum_{i\in J}\hat v_i^2g_i^2\Big)^{-p}=\frac1{\Gamma(p)}\int_0^\infty t^{p-1}\prod_{i\in J}\big(1+2t\hat v_i^2\big)^{-1/2}\dd t .
\]
Let $m_J=\max_{i\in J}\hat v_i^2\le\eta_v^2$.
The function $x\mapsto\log(1+2tx)$ is concave and lies above its chord on $[0,m_J]$, that is, $\log(1+2t\hat v_i^2)\ge(\hat v_i^2/m_J)\log(1+2tm_J)$ for $i\in J$.
Then the product is at most $(1+2tm_J)^{-a}$ with $a=q_J/(2m_J)$, and with the substitution $s=2tm_J$ and the beta integral we find
\[
\frac1{\Gamma(p)}\int_0^\infty t^{p-1}(1+2tm_J)^{-a}\dd t=\frac{\Gamma(a-p)}{(2m_J)^p\,\Gamma(a)}=\prod_{r=1}^p\big(q_J-2rm_J\big)^{-1},\qquad p=1,2 .
\]
Since $2pm_J\le4\eta_v^2\le1/12\le q_J/2$, each factor satisfies $q_J-2rm_J\ge q_J/2\ge1/12$; in particular $a>p$, and the integral converges.
This bounds the product by $12^p$ and proves \eqref{gp-pin-radial-inverse-bound-fourth}.
\end{proof}

\begin{lemma}[Integration by parts for quadratic statistics]\label{gp-pin-tangent-ibp-fourth}
Let $v$ be a finite variance sequence, and let $w\in\R^n$ satisfy $\sum_i\hat v_iw_i=0$.
Define, on $\mathcal S_{R^2}$,
\[
\begin{gathered}
Q_w=\sum_iw_i\big(Z_i^2-1\big),\qquad V=\sum_i\hat v_i^2Z_i^2,\qquad K_w=\sum_i\hat v_iw_iZ_i^2,\\
J_w=\sum_iw_i^2Z_i^2,\qquad L_w=\sum_i\hat v_i^2w_iZ_i^2,\qquad V_3=\sum_i\hat v_i^3Z_i^2,
\end{gathered}
\]
and
\begin{equation}\label{gp-pin-tangent-coefficients-fourth}
A_w=2J_w-\frac{2K_w^2}{V},\qquad R_w=\frac{2L_w}{V}-\frac{2K_wV_3}{V^2}.
\end{equation}
For every continuously differentiable $f:\R\to\R$,
\begin{equation}\label{gp-pin-tangent-identity-fourth}
\E_{\mu_v}\big[(Q_w+R_w)f(Q_w)\big]=\E_{\mu_v}\big[A_wf'(Q_w)\big].
\end{equation}
Further, there exist universal constants $\eta_{\mathrm{ibp}}>0$ and $C<\infty$ such that, if $\eta_v\le\eta_{\mathrm{ibp}}$, then
\begin{equation}\label{gp-pin-tangent-error-fourth}
\begin{gathered}
\E_{\mu_v}\big|A_w-2\|w\|_2^2\big|\le C\big(\|w\|_\infty\|w\|_2+\eta_v^2\|w\|_2^2\big),\\
\E_{\mu_v}|R_w|\le C\eta_v\|w\|_2,\qquad
\E_{\mu_v}|R_wQ_w|\le C\eta_v\|w\|_2^2 .
\end{gathered}
\end{equation}
\end{lemma}

The identity \eqref{gp-pin-tangent-identity-fourth} is exact and holds without any smallness assumption.
If $\mu_v$ were replaced by the law of independent Gaussian variables, the corresponding identity would hold with $R_w=0$ and $A_w=2J_w$; the term $-2K_w^2/V$ and the correction $R_w$ come from the constraint \eqref{eq:lik-constraint}.

\begin{proof}
We first prove the identity, using a vector field tangent to the spheres $\mathcal S_r$, and then bound the coefficients.

\paragraph{Step 1: a divergence identity on spheres.}
Let $X$ be a continuously differentiable vector field on $\R^n\setminus\{0\}$ with $X(y)\cdot y=0$ for all $y$.
We claim that $\int_{\mathcal S_r}\operatorname{div}X\dd\mathcal H^{n-1}=0$ for every $r>0$.
Indeed, for $0<a<b$, the divergence theorem on the shell $\{a\le|y|\le b\}$ gives $\int_{a\le|y|\le b}\operatorname{div}X\dd y=0$, since the outward normals at the two boundary spheres are $\pm y/|y|$ and $X\cdot y=0$ there.
In polar coordinates, this says that $\int_a^b\big(\int_{\mathcal S_{s^2}}\operatorname{div}X\dd\mathcal H^{n-1}\big)\dd s=0$ for all $0<a<b$.
The inner integral is continuous in $s$, and it vanishes for every $s>0$.

\paragraph{Step 2: the identity \eqref{gp-pin-tangent-identity-fourth}.}
We regard $z_i=y_i/\sqrt{v_i}$, and hence $Q_w,V,K_w,J_w,L_w,V_3,A_w,R_w$, as functions of $y\in\R^n$; they are smooth on $\R^n\setminus\{0\}$, where $V>0$.
Let $\alpha=K_w/V$ and define the vector field $\widetilde T$ on $\R^n\setminus\{0\}$ by
\[
\widetilde T_i(y)=\sqrt{v_i}\,\big(w_i-\alpha\hat v_i\big)z_i .
\]
Using $\sqrt{v_i}z_i=y_i$ and $v_i=S_v\hat v_i$, we have
\[
\widetilde T(y)\cdot y=S_v\sum_i\hat v_i(w_i-\alpha\hat v_i)z_i^2=S_v(K_w-\alpha V)=0 .
\]
Recalling that $\partial_{y_i}=v_i^{-1/2}\partial_{z_i}$, we find by direct computation that
\[
\begin{gathered}
\operatorname{div}\widetilde T=\sum_i(w_i-\alpha\hat v_i)-\sum_i\hat v_iz_i\partial_{z_i}\alpha,\qquad
\widetilde T\cdot\nabla\log\rho=-\sum_i(w_i-\alpha\hat v_i)z_i^2,\\
\widetilde T\cdot\nabla Q_w=2J_w-2\alpha K_w=A_w,
\end{gathered}
\]
where $\rho(y)=\exp\{-\frac12\sum_iy_i^2/v_i\}$.
Further, $\partial_{z_i}\alpha=2\hat v_iw_iz_i/V-2K_w\hat v_i^2z_i/V^2$, which gives $\sum_i\hat v_iz_i\partial_{z_i}\alpha=R_w$.
Adding the first two identities gives
\[
\rho^{-1}\operatorname{div}\big(\rho\widetilde T\big)=-\sum_i(w_i-\alpha\hat v_i)\big(z_i^2-1\big)-R_w=-Q_w+\alpha\sum_i\hat v_i\big(z_i^2-1\big)-R_w .
\]
On $\mathcal S_{R^2}$, the middle term vanishes by \eqref{eq:lik-constraint}.
Now apply Step~1 to $X=\rho f(Q_w)\widetilde T$, which is continuously differentiable on $\R^n\setminus\{0\}$ and satisfies $X\cdot y=0$.
Since $\operatorname{div}X=\rho\big[f(Q_w)\rho^{-1}\operatorname{div}(\rho\widetilde T)+f'(Q_w)\widetilde T\cdot\nabla Q_w\big]$, we obtain
\[
\int_{\mathcal S_{R^2}}\rho\big[-(Q_w+R_w)f(Q_w)+A_wf'(Q_w)\big]\dd\mathcal H^{n-1}=0 .
\]
The density of $\mu_v$ with respect to $\mathcal H^{n-1}$ on $\mathcal S_{R^2}$ is proportional to $\rho$, by \eqref{gp-pin-definition}, and this identity is \eqref{gp-pin-tangent-identity-fourth}.

\paragraph{Step 3: the error bounds.}
We take $\eta_{\mathrm{ibp}}\le\min\{\eta_0,\eta_{\mathrm{inv}}\}$, where $\eta_0$ is the constant of \cref{gp-quadratic-lp} and $\eta_{\mathrm{inv}}$ that of \cref{gp-pin-radial-inverse-fourth}.
All norms below are in $L^p(\mu_v)$.
We have $\|Q_w\|_4\le C\|w\|_2$ by \cref{gp-quadratic-lp}.
Since $\sum_i\hat v_iw_i=0$, we have $K_w=\sum_i\hat v_iw_i(Z_i^2-1)$, and $\|K_w\|_4\le C\eta_v\|w\|_2$ by \cref{gp-quadratic-lp} with coefficients $\hat v_iw_i$.
Similarly, using $J_w-\|w\|_2^2=\sum_iw_i^2(Z_i^2-1)$, we find that
\[
\big\|J_w-\|w\|_2^2\big\|_1\le\big\|J_w-\|w\|_2^2\big\|_2\le C\Big(\sum_iw_i^4\Big)^{1/2}\le C\|w\|_\infty\|w\|_2 .
\]
Next, $L_w=\sum_i\hat v_i^2w_i+\sum_i\hat v_i^2w_i(Z_i^2-1)$.
By the Cauchy--Schwarz inequality, $|\sum_i\hat v_i^2w_i|\le(\sum_i\hat v_i^2)^{1/2}(\sum_i\hat v_i^2w_i^2)^{1/2}\le\eta_v\|w\|_2$, and \cref{gp-quadratic-lp} bounds the second sum in $L^4$ by $C(\sum_i\hat v_i^4w_i^2)^{1/2}\le C\eta_v^2\|w\|_2$.
We conclude that $\|L_w\|_4\le C\eta_v\|w\|_2$.
Finally, $0\le V_3\le\eta_vV$ pointwise, so that $|R_w|\le2(|L_w|+\eta_v|K_w|)/V$, and $\|V^{-1}\|_2\le C$ by \cref{gp-pin-radial-inverse-fourth}.

We now combine these bounds.
By H\"older's inequality,
\[
\E\big|A_w-2\|w\|_2^2\big|\le2\big\|J_w-\|w\|_2^2\big\|_1+2\|K_w\|_4^2\|V^{-1}\|_2\le C\big(\|w\|_\infty\|w\|_2+\eta_v^2\|w\|_2^2\big).
\]
Using the Cauchy--Schwarz inequality, we find
\[
\E|R_w|\le2\big(\|L_w\|_2+\eta_v\|K_w\|_2\big)\|V^{-1}\|_2\le C\eta_v\|w\|_2,
\]
and by H\"older's inequality with exponents $4$, $4$, and $2$,
\[
\E|R_wQ_w|\le2\big(\|L_w\|_4+\eta_v\|K_w\|_4\big)\|Q_w\|_4\|V^{-1}\|_2\le C\eta_v\|w\|_2^2 .
\]
This proves \eqref{gp-pin-tangent-error-fourth}.
\end{proof}

\section{Normal approximation of quadratic statistics}\label{sec:lik-normal}

\begin{lemma}[Solutions of the Stein equation]\label{gp-scalar-stein-tests-fourth}
There exists a universal constant $C$ with the following property.
Let $h:\R\to\R$ be smooth, and suppose that $h$ is $1$-Lipschitz or that $\|h\|_\infty\le1$.
Then there exists a smooth bounded solution $f$ of
\begin{equation}\label{gp-scalar-stein-equation-fourth}
f'(x)-xf(x)=h(x)-\E h(G),\qquad x\in\R,
\end{equation}
with $\|f\|_\infty+\|f'\|_\infty\le C$.
\end{lemma}

For background on Stein's equation and the generator approach, see \cite{Stein1972} and \cite[Chapter~2]{ChenGoldsteinShao2011}. We prove the bounds here.

\begin{proof}
We set $\bar h=h-\E h(G)$ and define
\[
f(x)=e^{x^2/2}\int_{-\infty}^x\bar h(u)e^{-u^2/2}\dd u=-e^{x^2/2}\int_x^\infty\bar h(u)e^{-u^2/2}\dd u,
\]
where the two expressions agree since $\int\bar h\phi=0$.
Differentiation shows that $f$ is smooth and solves \eqref{gp-scalar-stein-equation-fourth}.
It is the only bounded solution, since the solutions of the homogeneous equation are the multiples of $e^{x^2/2}$.

Suppose $\|h\|_\infty\le1$, so that $|\bar h|\le2$.
For $x>0$, the substitution $u=x+s$ shows that
\[
e^{x^2/2}\int_x^\infty e^{-u^2/2}\dd u=\int_0^\infty e^{-xs-s^2/2}\dd s\le\min\big\{\sqrt{\pi/2},\,1/x\big\},
\]
and the second expression for $f$ gives $|f(x)|\le2\sqrt{\pi/2}$ and $|xf(x)|\le2$.
The case $x<0$ follows from the first expression in the same way, and the bounds at $x=0$ follow by continuity.
Finally, \eqref{gp-scalar-stein-equation-fourth} gives $|f'|\le|xf|+|\bar h|\le4$.

Suppose $h$ is $1$-Lipschitz, and assume first that all positive-order derivatives of $h$ are bounded.
Let $P_th(x)=\E h(e^{-t}x+s_tG)$ with $s_t=(1-e^{-2t})^{1/2}$ be the Ornstein--Uhlenbeck semigroup, and let
\[
F(x)=-\int_0^\infty e^{-t}P_t(h')(x)\dd t .
\]
Then $\|F\|_\infty\le\|h'\|_\infty\le1$.
For $t>0$, Gaussian integration by parts gives $(P_t(h'))'(x)=e^{-t}\E[h''(e^{-t}x+s_tG)]=e^{-t}s_t^{-1}\E[Gh'(e^{-t}x+s_tG)]$, and
\[
|F'(x)|\le\int_0^\infty\frac{e^{-2t}}{s_t}\,\E|G|\dd t=\E|G|=\sqrt{2/\pi},
\]
where we used $\int_0^\infty e^{-2t}(1-e^{-2t})^{-1/2}\dd t=1$.
Further, $(P_th)'=e^{-t}P_t(h')$, and $\partial_tP_th=(P_th)''-x(P_th)'$ by direct differentiation of the Gaussian integral.
We obtain
\[
F'(x)-xF(x)=-\int_0^\infty\big[(P_th)''(x)-x(P_th)'(x)\big]\dd t=-\int_0^\infty\partial_tP_th(x)\dd t=h(x)-\E h(G),
\]
since $P_0h=h$ and $P_th(x)\to\E h(G)$ as $t\to\infty$.
The bounds on the integrands justify the interchanges of derivatives and integrals.
We conclude that $F$ is a bounded solution of \eqref{gp-scalar-stein-equation-fourth}, and $F=f$.
For a general smooth $1$-Lipschitz $h$, let $h_\varepsilon=h*\rho_\varepsilon$ for a nonnegative smooth compactly supported mollifier of unit mass. Then $h_\varepsilon$ remains $1$-Lipschitz, has bounded derivatives of every positive order, and converges uniformly to $h$. The integral formula gives $f_\varepsilon(x)\to f(x)$, while $f_\varepsilon'(x)=xf_\varepsilon(x)+h_\varepsilon(x)-\E h_\varepsilon(G)\to f'(x)$ for each $x$. Thus $\|f\|_\infty\le1$ and $\|f'\|_\infty\le\sqrt{2/\pi}$.
\end{proof}

\begin{theorem}[Normal approximation of quadratic statistics]\label{gp-conditional-normal-third}
There exist universal constants $\eta_{\mathrm{nor}}>0$ and $C<\infty$ with the following property.
Let $v$ be a finite variance sequence with $\eta_v\le\eta_{\mathrm{nor}}$, and let $w\in\R^n$ be arbitrary.
Let
\[
w^\perp=w-2\langle\hat v,w\rangle\hat v,\qquad \sigma_w^2=2\|w^\perp\|_2^2,\qquad Q_w=\sum_iw_i\big(Z_i^2-1\big).
\]
If $\sigma_w=0$, then $Q_w=0$ on $\mathcal S_{R^2}$.
In all cases,
\begin{equation}\label{gp-conditional-normal-w1-third}
W_1\big(\Law_{\mu_v}(Q_w),\cN(0,\sigma_w^2)\big)\le C\big(\eta_v\sigma_w+\|w^\perp\|_\infty\big).
\end{equation}
If $\sigma_w>0$, then also
\begin{equation}\label{gp-conditional-normal-tv-third}
\big\|\Law_{\mu_v}(Q_w)-\cN(0,\sigma_w^2)\big\|_{\TV}\le\min\big\{1,C\big(\eta_v+\|w^\perp\|_\infty/\sigma_w\big)\big\}.
\end{equation}
Further,
\begin{align}
\big|\E_{\mu_v}Q_w\big|&\le C\big(\eta_v\sigma_w+\|w^\perp\|_\infty\big),\label{gp-conditional-normal-mean-third}\\
\big|\E_{\mu_v}Q_w^2-\sigma_w^2\big|&\le C\big(\eta_v\sigma_w^2+\sigma_w\|w^\perp\|_\infty\big).\label{gp-conditional-normal-second-third}
\end{align}
\end{theorem}

Since $\|\hat v\|_2^2=1/2$, the vector $w^\perp$ is the orthogonal projection of $w$ onto the orthogonal complement of $\hat v$.
The variance $\sigma_w^2$ is therefore no greater than the variance $2\|w\|_2^2$ of $\sum_iw_i(g_i^2-1)$ for independent Gaussian variables, by the part of $w$ along the constrained direction.
The constants do not depend on the dimension or on the smallest variance.

\begin{proof}
We set $\eta_{\mathrm{nor}}=\eta_{\mathrm{ibp}}$, the constant of \cref{gp-pin-tangent-ibp-fourth}.
By \eqref{eq:lik-constraint}, we have $Q_w-Q_{w^\perp}=2\langle\hat v,w\rangle\sum_i\hat v_i(Z_i^2-1)=0$, which means that $Q_w=Q_{w^\perp}$ on $\mathcal S_{R^2}$; also $\langle\hat v,w^\perp\rangle=0$.
If $\sigma_w=0$, then $w^\perp=0$ and $Q_w=0$.
Let $\sigma_w>0$ and $u=w^\perp/\sigma_w$, so that $\langle\hat v,u\rangle=0$, $2\|u\|_2^2=1$, and $Q_u=Q_w/\sigma_w$.

Let $h$ be smooth, and either $1$-Lipschitz with bounded derivatives of every positive order or bounded by one, and let $f$ be the solution of \eqref{gp-scalar-stein-equation-fourth} from \cref{gp-scalar-stein-tests-fourth}.
By \eqref{gp-pin-tangent-identity-fourth} for $u$, we have $\E[Q_uf(Q_u)]=\E[A_uf'(Q_u)]-\E[R_uf(Q_u)]$, and
\[
\E h(Q_u)-\E h(G)=\E\big[f'(Q_u)-Q_uf(Q_u)\big]=\E\big[(1-A_u)f'(Q_u)+R_uf(Q_u)\big].
\]
Since $1=2\|u\|_2^2$, the bounds \eqref{gp-pin-tangent-error-fourth} imply that
\begin{equation}\label{eq:lik-stein-bound}
\big|\E h(Q_u)-\E h(G)\big|\le C\big(\E|A_u-2\|u\|_2^2|+\E|R_u|\big)\le C\big(\eta_v+\|w^\perp\|_\infty/\sigma_w\big),
\end{equation}
where we used $\|u\|_\infty\|u\|_2\le\|w^\perp\|_\infty/\sigma_w$ and $\eta_v^2\le\eta_v$.

Let $h$ be $1$-Lipschitz.
The convolution $h_\delta$ of $h$ with a smooth probability density supported in $[-\delta,\delta]$ is smooth, $1$-Lipschitz, has bounded derivatives of every positive order, and satisfies $\|h_\delta-h\|_\infty\le\delta$.
Applying \eqref{eq:lik-stein-bound} to $h_\delta$ and letting $\delta\to0$, we find that \eqref{eq:lik-stein-bound} holds for $h$.
Taking the supremum over $h$, we obtain $W_1(\Law(Q_u),\cN(0,1))\le C(\eta_v+\|w^\perp\|_\infty/\sigma_w)$.
Since $W_1(\Law(\sigma X),\Law(\sigma X'))=\sigma W_1(\Law(X),\Law(X'))$ for $\sigma>0$, this is \eqref{gp-conditional-normal-w1-third}.

For \eqref{gp-conditional-normal-tv-third}, we claim that the total variation distance between two probability measures $\lambda_1,\lambda_2$ on $\R$ is the supremum of $|\int h\dd\lambda_1-\int h\dd\lambda_2|$ over smooth $h$ with $0\le h\le1$.
Indeed, given a Borel set $E$ and $\delta>0$, by regularity of the finite measure $\lambda_1+\lambda_2$ there exist a compact $K$ and an open $O$ with $K\subset E\subset O$ and $(\lambda_1+\lambda_2)(O\setminus K)\le\delta$.
Covering $K$ by finitely many open intervals whose closures lie in $O$, we find a smooth $h$ with $\ind_K\le h\le\ind_O$, and then $|\lambda_j(E)-\int h\dd\lambda_j|\le\delta$ for $j=1,2$.
By the claim and \eqref{eq:lik-stein-bound} for smooth $h$ with values in $[0,1]$, we have $\|\Law(Q_u)-\cN(0,1)\|_{\TV}\le C(\eta_v+\|w^\perp\|_\infty/\sigma_w)$.
Total variation is invariant under multiplication of both variables by $\sigma_w$, and it is at most one, which proves \eqref{gp-conditional-normal-tv-third}.

For the moment bounds, we apply \eqref{gp-pin-tangent-identity-fourth} to $w^\perp$ with $f(x)=1$ and $f(x)=x$, which gives
\[
\E Q_{w^\perp}=-\E R_{w^\perp},\qquad \E Q_{w^\perp}^2=\E A_{w^\perp}-\E\big[R_{w^\perp}Q_{w^\perp}\big].
\]
Using $Q_w=Q_{w^\perp}$, $2\|w^\perp\|_2^2=\sigma_w^2$, and \eqref{gp-pin-tangent-error-fourth}, we have $|\E Q_w|\le C\eta_v\|w^\perp\|_2\le C\eta_v\sigma_w$, which implies \eqref{gp-conditional-normal-mean-third}, and
\[
\big|\E Q_w^2-\sigma_w^2\big|\le\E\big|A_{w^\perp}-2\|w^\perp\|_2^2\big|+\E|R_{w^\perp}Q_{w^\perp}|
\le C\big(\|w^\perp\|_\infty\sigma_w+\eta_v\sigma_w^2\big),
\]
which is \eqref{gp-conditional-normal-second-third}.
\end{proof}

\section{Likelihood ratios}\label{sec:lik-lan}

Let $v$ and $\tilde v$ be finite variance sequences on $\{1,\dots,n\}$ with $\sum_iv_i=\sum_i\tilde v_i=R^2$.
Then $\mu_v$ and $\mu_{\tilde v}$ are laws on the same sphere $\mathcal S_{R^2}$, with positive continuous densities, so they are mutually absolutely continuous.
We write
\begin{equation}\label{eq:lik-perturbation}
\tilde v_i=v_i(1+e_i),\qquad \tilde e_i=\frac{e_i}{1+e_i},\qquad \bar e=\max_i|e_i|,\qquad I=\frac12\sum_ie_i^2,
\end{equation}
where $I=\cI(v,\tilde v)$ is the information of $\tilde v$ relative to $v$ (\cref{def:model-information}).
The equality of the traces means that $\sum_iv_ie_i=0$, that is,
\begin{equation}\label{eq:lik-trace}
\sum_i\hat v_ie_i=0 .
\end{equation}
By \eqref{eq:cond-normalization}, $\dd\mu_v/\dd\mathcal H^{n-1}=\gamma_v/(2Rp_v(0))$ on $\mathcal S_{R^2}$, and $\gamma_{\tilde v}(y)/\gamma_v(y)=\prod_i(1+e_i)^{-1/2}\exp\{\frac12\sum_i\tilde e_iy_i^2/v_i\}$, since $v_i^{-1}-\tilde v_i^{-1}=\tilde e_i/v_i$.
We conclude that the log likelihood ratio $\ell=\log(\dd\mu_{\tilde v}/\dd\mu_v)$ is, exactly,
\begin{equation}\label{gp-lan-score-center-third}
\ell=k_0+Q_w,\qquad w=\tfrac12\tilde e,\qquad
k_0=\frac12\sum_i\big[\tilde e_i-\log(1+e_i)\big]+\log\frac{p_v(0)}{p_{\tilde v}(0)},
\end{equation}
where $Q_w=\sum_iw_i(Z_i^2-1)$ as in \cref{gp-conditional-normal-third}.
Equivalently, with $\ell^{\mathrm{prod}}=-\frac12\sum_i\log(1+e_i)+\frac12\sum_i\tilde e_iZ_i^2$ the log likelihood ratio of the unconditioned Gaussian laws,
\begin{equation}\label{gp-lan-exact-likelihood}
e^{\ell}=e^{\ell^{\mathrm{prod}}}\,\frac{p_v(0)}{p_{\tilde v}(0)} .
\end{equation}

\begin{theorem}[Quantitative likelihood ratios on a common sphere]\label{gp-pinned-lan-quantitative-third}
There exists a universal constant $\eta_{\mathrm{lik}}>0$ such that, for every $B<\infty$, there exists $C_B<\infty$ with the following property.
Let $v,\tilde v$ be finite variance sequences with $\sum_iv_i=\sum_i\tilde v_i$, and define $e_i,\bar e,I$ by \eqref{eq:lik-perturbation}.
Suppose that
\[
\bar e\le\frac14,\qquad \eta_v\le\eta_{\mathrm{lik}},\qquad I\le B .
\]
Then, with $\ell=\log(\dd\mu_{\tilde v}/\dd\mu_v)$,
\begin{equation}\label{gp-pinned-lan-w1-third}
W_1\big(\Law_{\mu_v}(\ell),\cN(-I/2,I)\big)\le C_B\big(\eta_v+\bar e\big),
\end{equation}
and
\begin{equation}\label{gp-pinned-lan-profile-error-third}
\Big|\big\|\mu_{\tilde v}-\mu_v\big\|_{\TV}-\big(2\Phi(\sqrt I/2)-1\big)\Big|\le C_B\big(\eta_v+\bar e\big).
\end{equation}
\end{theorem}

The threshold $\eta_{\mathrm{lik}}$ does not depend on $B$, and the bounds hold uniformly down to $I=0$.
The constants do not depend on the dimension or on the smallest variance.

\begin{proof}
If $I=0$, then $\tilde v=v$, $\ell=0$, and the left sides of \eqref{gp-pinned-lan-w1-third} and \eqref{gp-pinned-lan-profile-error-third} vanish, so both inequalities hold.
Let $I>0$.
We use the exact decomposition \eqref{gp-lan-score-center-third}: we show that the constant $k_0$ is close to $-I/2$, that the projected coefficient vector $w^\perp$ of \cref{gp-conditional-normal-third} has small entries and norm close to $\sqrt{I/2}$, and then apply \cref{gp-conditional-normal-third,lem:lik-tv}.
We take $\eta_{\mathrm{lik}}\le\min\{\frac35\eta_{\mathrm{loc}},\eta_{\mathrm{nor}}\}$, where $\eta_{\mathrm{loc}}$ is the constant of \cref{gp-local-limit} and $\eta_{\mathrm{nor}}$ that of \cref{gp-conditional-normal-third}.
Note that $|\tilde e_i|\le|e_i|/(1-\bar e)\le\frac43|e_i|$.

\paragraph{Step 1: the constant $k_0$.}
Recall that $S_{\tilde v}^2=2\sum_iv_i^2(1+e_i)^2$, so that $1-\bar e\le S_{\tilde v}/S_v\le1+\bar e$ and $\eta_{\tilde v}\le\frac{1+\bar e}{1-\bar e}\eta_v\le\frac53\eta_v\le\eta_{\mathrm{loc}}$.
Writing $p_v(0)=f_v(0)/S_v$, and similarly for $\tilde v$, we have
\[
\log\frac{p_v(0)}{p_{\tilde v}(0)}=\log\frac{f_v(0)}{f_{\tilde v}(0)}+\log\frac{S_{\tilde v}}{S_v}.
\]
By \cref{gp-local-limit}, $f_v(0)$ and $f_{\tilde v}(0)$ are at least $\phi(0)/2$ and differ from $\phi(0)$ by at most $C\eta_v$, and the first logarithm is at most $C\eta_v$ in absolute value; the second is at most $\frac43\bar e$.
Thus
\begin{equation}\label{gp-lan-normalizer}
\Big|\log\frac{p_v(0)}{p_{\tilde v}(0)}\Big|\le C\big(\eta_v+\bar e\big).
\end{equation}
Next, the function $\kappa(e)=\frac12[e/(1+e)-\log(1+e)]+e^2/4$ satisfies $\kappa(0)=\kappa'(0)=\kappa''(0)=0$ and $\kappa'''(e)=(2-e)(1+e)^{-4}$, which is at most $8$ for $|e|\le1/4$.
Taylor's formula gives $|\kappa(e)|\le\frac43|e|^3$ for $|e|\le1/4$. Summing over $i$, we obtain
\[
\Big|\frac12\sum_i\big[\tilde e_i-\log(1+e_i)\big]+\frac I2\Big|\le\frac43\sum_i|e_i|^3\le\frac83\bar eI .
\]
We conclude that
\begin{equation}\label{eq:lik-center}
|k_0+I/2|\le C\big(\eta_v+\bar e\big)+3\bar eI\le C_B\big(\eta_v+\bar e\big).
\end{equation}

\paragraph{Step 2: the projected coefficients.}
With $w=\frac12\tilde e$, we have $2\langle\hat v,w\rangle=\langle\hat v,\tilde e\rangle$ and $w^\perp=\frac12\tilde e-\langle\hat v,\tilde e\rangle\hat v$.
By \eqref{eq:lik-trace} and $\tilde e_i-e_i=-e_i^2/(1+e_i)$,
\[
\big|\langle\hat v,\tilde e\rangle\big|=\Big|\sum_i\hat v_i\frac{e_i^2}{1+e_i}\Big|\le\frac43\eta_v\sum_ie_i^2=\frac83\eta_vI .
\]
This cancellation, which comes from the equality of the traces, keeps the projection small, and it gives
\[
\|w^\perp\|_\infty\le\frac23\bar e+\frac83\eta_v^2I .
\]
Recall that $w^\perp$ is the orthogonal projection of $w$ onto the complement of $\hat v$ and $\|\hat v\|_2^2=1/2$, so that
\[
\sigma_w^2=2\|w^\perp\|_2^2=\frac12\|\tilde e\|_2^2-\langle\hat v,\tilde e\rangle^2 .
\]
Using $|\tilde e_i^2-e_i^2|=e_i^2|(1+e_i)^{-2}-1|\le4\bar ee_i^2$, we find
\[
\big|\sigma_w^2-I\big|\le4\bar eI+8\eta_v^2I^2,\qquad \sigma_w^2\le\frac12\|\tilde e\|_2^2\le2I .
\]
Dividing by $\sigma_w+\sqrt I\ge\sqrt I$, we find
\begin{equation}\label{eq:lik-sigma}
\big|\sigma_w-\sqrt I\big|\le4\bar e\sqrt I+8\eta_v^2I^{3/2}.
\end{equation}

\paragraph{Step 3: conclusion.}
By \cref{gp-conditional-normal-third} and Step~2,
\[
W_1\big(\Law_{\mu_v}(Q_w),\cN(0,\sigma_w^2)\big)\le C\big(\eta_v\sigma_w+\|w^\perp\|_\infty\big)\le C_B\big(\eta_v+\bar e\big).
\]
Coupling $\cN(0,\sigma_w^2)$ and $\cN(0,I)$ as $\sigma_wG$ and $\sqrt IG$, and using \eqref{eq:lik-sigma}, we have $W_1(\cN(0,\sigma_w^2),\cN(0,I))\le\E|G|\,|\sigma_w-\sqrt I|\le C_B(\eta_v+\bar e)$.
Since $\ell=k_0+Q_w$ and a deterministic shift changes $W_1$ by at most its size, \eqref{eq:lik-center} implies that
\[
W_1\big(\Law_{\mu_v}(\ell),\cN(-I/2,I)\big)\le|k_0+I/2|+W_1\big(\Law_{\mu_v}(Q_w),\cN(0,I)\big)\le C_B\big(\eta_v+\bar e\big),
\]
which is \eqref{gp-pinned-lan-w1-third}.
Finally, by \cref{lem:lik-tv} with $P=\mu_v$ and $Q=\mu_{\tilde v}$, we have $\|\mu_{\tilde v}-\mu_v\|_{\TV}=\E_{\mu_v}(1-e^\ell)_+$, and the last statement of \cref{lem:lik-tv}, together with \eqref{eq:lik-gaussian-profile}, proves \eqref{gp-pinned-lan-profile-error-third}.
\end{proof}

\begin{remark}[Disorder-dependent sequences]\label{rem:lik-random}
\Cref{gp-pinned-lan-quantitative-third} is deterministic.
In the applications, $v$ and $\tilde v$ depend on the disorder and on time, and the hypotheses are verified on events of high probability.
The bound $B$ is then fixed before $N\to\infty$: if $\eta_v+\bar e\to0$ in probability and the information is tight, then for each $\delta>0$ we choose a deterministic $B$ with $\limsup_N\PP(I>B)\le\delta$, apply \eqref{gp-pinned-lan-profile-error-third} on the event $\{I\le B,\ \bar e\le1/4,\ \eta_v\le\eta_{\mathrm{lik}}\}$, and then let $\delta\to0$.
We obtain $\|\mu_{\tilde v}-\mu_v\|_{\TV}-(2\Phi(\sqrt I/2)-1)\to0$ in probability.
The statement concerns a fixed pair of laws; uniformity over a family of pairs indexed by time requires the hypotheses to hold uniformly over the family.
\end{remark}

\begin{theorem}[Gaussian limits of likelihood ratios on a common sphere]\label{gp-pinned-lan}
For each $N\ge1$, let $v=v^{(N)}$ and $\tilde v=\tilde v^{(N)}$ be finite variance sequences with
\begin{equation}\label{gp-lan-assumptions}
\sum_iv_i=\sum_i\tilde v_i,\qquad \tilde v_i=v_i(1+e_i),\qquad \bar e_N=\max_i|e_i|\to0,\qquad \eta_v\to0,
\end{equation}
as $N\to\infty$, where the indices $i$ and the coefficients $e_i$ depend on $N$.
Let $I_N=\frac12\sum_ie_i^2$ and $\ell_N=\log(\dd\mu_{\tilde v}/\dd\mu_v)$.
If $I_N\to I\in[0,\infty)$, then under $\mu_v$,
\begin{equation}\label{gp-lan-mgf}
\ell_N\Rightarrow\cN(-I/2,I),\qquad \E_{\mu_v}e^{t\ell_N}\to e^{(t^2-t)I/2}\quad\text{for each fixed }t\in\R,
\end{equation}
and
\begin{equation}\label{gp-lan-tv}
\big\|\mu_{\tilde v}-\mu_v\big\|_{\TV}\to2\Phi\big(\sqrt I/2\big)-1 .
\end{equation}
If $I_N\to\infty$, then $\|\mu_{\tilde v}-\mu_v\|_{\TV}\to1$.
\end{theorem}

The convergence of exponential moments is for each fixed $t$; in its proof, the tilted variances are positive only for $N$ large depending on $t$.
In the case $I_N\to\infty$, no relation between $\eta_v$ and $I_N$ is assumed.

\begin{proof}
We prove the three assertions in turn.

\paragraph{Weak convergence and total variation.}
Let $I_N\to I<\infty$ and $B=I+1$.
For $N$ large, $\bar e_N\le1/4$, $\eta_v\le\eta_{\mathrm{lik}}$, and $I_N\le B$, and by \cref{gp-pinned-lan-quantitative-third} we have $W_1(\Law_{\mu_v}(\ell_N),\cN(-I_N/2,I_N))\le C_B(\eta_v+\bar e_N)\to0$.
Coupling with one standard Gaussian variable, we have $W_1(\cN(-I_N/2,I_N),\cN(-I/2,I))\le|I_N-I|/2+\E|G|\,|\sqrt{I_N}-\sqrt I|\to0$.
Convergence in $W_1$ implies weak convergence, which proves the first part of \eqref{gp-lan-mgf}, including the case $I=0$.
By \eqref{gp-pinned-lan-profile-error-third} and continuity of $I\mapsto2\Phi(\sqrt I/2)-1$, we obtain \eqref{gp-lan-tv}.

\paragraph{Exponential moments.}
Fix $t\in\R$.
Recall that $|\tilde e_i|\le\frac43\bar e_N\to0$; we have $|t\tilde e_i|\le1/2$ and $|(1-t)e_i|\le1/2$ for all $i$ when $N$ is large. We assume this from now on.
By \eqref{gp-lan-exact-likelihood},
\[
\E_{\mu_v}e^{t\ell_N}=\Big(\frac{p_v(0)}{p_{\tilde v}(0)}\Big)^t\prod_i(1+e_i)^{-t/2}\,\E_{\mu_v}\exp\Big\{\frac t2\sum_i\tilde e_iZ_i^2\Big\}.
\]
We apply \cref{gp-real-tilt} with coefficients $\tilde e_i$.
The tilted variances are
\begin{equation}\label{gp-lan-tilt}
v_i(t)=\frac{v_i}{1-t\tilde e_i}=\frac{v_i(1+e_i)}{1+(1-t)e_i},\qquad
K_N(t)=\prod_i(1+e_i)^{-t/2}(1-t\tilde e_i)^{-1/2},
\end{equation}
and we obtain
\[
\E_{\mu_v}e^{t\ell_N}=\Big(\frac{p_v(0)}{p_{\tilde v}(0)}\Big)^tK_N(t)\,\frac{p_{v(t)}(-\Delta_N(t))}{p_v(0)},\qquad \Delta_N(t)=\sum_i\big(v_i(t)-v_i\big).
\]
We treat the three factors separately.
The first tends to one, by \eqref{gp-lan-normalizer}.
For the third, using the identity $\sum_iv_ie_i=0$, we find, exactly,
\begin{equation}\label{gp-lan-radial-shift}
\Delta_N(t)=\sum_i\frac{tv_ie_i}{1+(1-t)e_i}=-t(1-t)\sum_i\frac{v_ie_i^2}{1+(1-t)e_i},\quad
|\Delta_N(t)|\le2|t(1-t)|\,\eta_vS_v\sum_ie_i^2 .
\end{equation}
The ratios $v_i(t)/v_i$ tend to one uniformly in $i$, so that $S_{v(t)}/S_v\to1$ and $\eta_{v(t)}\le\eta_v(1+o(1))\to0$.
As $I_N$ is bounded, we deduce from \eqref{gp-lan-radial-shift} that $\Delta_N(t)/S_{v(t)}\to0$.
Using \cref{gp-local-limit} with $j=0$, we find that $f_{v(t)}\to\phi$ and $f_v\to\phi$ uniformly. Since $\phi$ is continuous, it follows that
\begin{equation}\label{gp-lan-density-ratio}
\frac{p_{v(t)}(-\Delta_N(t))}{p_v(0)}=\frac{S_v}{S_{v(t)}}\,\frac{f_{v(t)}(-\Delta_N(t)/S_{v(t)})}{f_v(0)}\to1 .
\end{equation}
For the second factor, $\log K_N(t)=\sum_ik_t(e_i)$ with
\[
k_t(e)=\frac{1-t}2\log(1+e)-\frac12\log\big(1+(1-t)e\big),
\]
which follows from $1-t\tilde e_i=(1+(1-t)e_i)/(1+e_i)$.
We have $k_t(0)=k_t'(0)=0$ and $k_t''(0)=(t^2-t)/2$, and Taylor's formula gives $|k_t(e)-(t^2-t)e^2/4|\le C_t|e|^3$ for $|e|$ small, depending on $t$.
We obtain
\[
\log K_N(t)=\frac{t^2-t}4\sum_ie_i^2+O_t\Big(\bar e_N\sum_ie_i^2\Big)\to\frac{(t^2-t)I}2 .
\]
Combining the three factors, we obtain the second part of \eqref{gp-lan-mgf}.

\paragraph{Diverging information.}
Let $I_N\to\infty$, and consider the single quadratic statistic
\[
Q_N(y)=\sum_ie_i\Big(\frac{y_i^2}{v_i}-1\Big).
\]
For $N$ large, $\eta_v$ and $\eta_{\tilde v}\le\frac53\eta_v$ are at most the constant $\eta_0$ of \cref{gp-quadratic-lp}.
Under $\mu_v$, by \cref{gp-quadratic-lp} with $p=2$, we have $\E_{\mu_v}Q_N^2\le C\sum_ie_i^2=2CI_N$.
Under $\mu_{\tilde v}$, with $\tilde Z_i=Y_i/\sqrt{\tilde v_i}$, we have $y_i^2/v_i=(1+e_i)\tilde Z_i^2$, and hence the exact identity
\[
Q_N=2I_N+\sum_ie_i(1+e_i)\big(\tilde Z_i^2-1\big).
\]
By \cref{gp-quadratic-lp} under $\mu_{\tilde v}$, we have $\E_{\mu_{\tilde v}}(Q_N-2I_N)^2\le C\sum_ie_i^2(1+e_i)^2\le CI_N$.
By Chebyshev's inequality,
\[
\mu_v\big(Q_N>I_N\big)\le\frac{C}{I_N},\qquad \mu_{\tilde v}\big(Q_N\le I_N\big)\le\mu_{\tilde v}\big(|Q_N-2I_N|\ge I_N\big)\le\frac{C}{I_N}.
\]
We conclude that $\|\mu_{\tilde v}-\mu_v\|_{\TV}\ge\mu_{\tilde v}(Q_N>I_N)-\mu_v(Q_N>I_N)\ge1-2C/I_N\to1$, as desired.
\end{proof}