Normal distribution by successive approximation?

Question

$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see e.g. this answer.

This appears to make the conjecture below somewhat plausible.

Let $\mu$ be any probability measure on $\R^2$ with a finite nonzero covariance matrix. Let $\mu_1:=\mu$. For each natural $n$, consider the following three-step procedure:

1. let $$\la_n:=\mu_n^{(1)}\otimes\mu_n^{(2)},$$ where $\mu_n^{(1)}$ and $\mu_n^{(2)}$ are the marginals of $\mu_n$;
2. let $$\nu_n:=\frac1{2\pi}\int_0^{2\pi}\la_n R_t\,dt,$$ where $\la_n R_t$ is the pushforward measure obtained from $\la_n$ by the rotation about the origin through angle $-t$;
3. let $\mu_{n+1}$ be obtained by rescaling the probability measure $\nu_n$ so that the covariance matrix of $\mu_{n+1}$ be the unit matrix.

Conjecture: $\mu_n$ converges weakly (as $n\to\infty$) to the standard Gaussian measure on $\R^2$.

Is this conjecture true?

Comment: Perhaps Step 3, the rescaling, is not essential. Of course, if we have the convergence to a (nondegenerate) Gaussian measure without Step 3, then we have such a convergence with Step 3 as well.

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We can restate the problem (without rescaling) analytically as follows. Let $f_n$ denote the characteristic function of $\nu_n$, so that $$f_n(u,v)=\int_{\R^2}\nu_n(dx\times dy)e^{i(ux+uy)} \\ =\int_{\R^2}\nu_n(dx\times dy)\cos(ux+uy)\quad \text{(by symmetry)}$$ for all real $u$ and $v$. Then $$f_n(u,v)=g_n\big(\sqrt{u^2+v^2}\big)$$ for some function $g_n\colon[0,\infty)\to\mathbb R$ and all real $u$ and $v$. Then for all natural $n$ and all real $r\ge0$ $$g_{n+1}(r)=\frac2\pi\int_0^{\pi/2} dt\,g_n(r\cos t)g_n(r\sin t). \tag{1}$$ We want to show that $g_n(r)\to e^{-c^2 r^2/2}$ for some $c\in(0,\infty)$ and all real $r\ge0$.

So, analytically, the problem may be viewed as one of stability of a (nonlinear) integral equation or as one of solving such an integral equation by iterations.

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Using substitutions $g_n(r)=h_n(r^2)$ and $r^2=s$, we can rewrite (1) as $$h_{n+1}(s)=\frac1\pi\int_0^s du\,\frac{h_n(s-u)}{\sqrt{s-u}}\frac{h_n(u)}{\sqrt{u}}$$ and then as $$\pi H_{n+1}(s)\sqrt s=[(H_n*H_n)(s)=]\int_0^s du\,H_n(s-u)H_n(u)$$ for all natural $n$ and all real $s\ge0$, where $H_n(u):=h_n(u)/\sqrt u$. We want to show that $H_n(u)\to e^{-c^2 u/2}/\sqrt u$ for some $c\in(0,\infty)$ and all real $u>0$.

Answer 1

In the rewritten form it doesn't require any ingenuity at all. For brevity, introduce the notation $$ (Th)(s)=\frac 1\pi\int_0^s\frac{h(s-u)h(u)}{\sqrt{u(s-u)}}\,du=\frac 1\pi\int_0^1\frac{h(s(1-v))h(sv)}{\sqrt{v(1-v)}}\,dv $$ Suppose that $h_0$ is any real valued Lipschitz function on $[0,+\infty)$ (the finiteness of the second moment guarantees Lipschitzness for your $h_0$ and, if you really want to discuss it, I can show how this particular condition can be dropped) such that $\|h_0\|_\infty=h_0(0)=1$ and there exists $h_0'(0)=-c< 0$ (the latter two conditions are essential). Then the iterations $h_{n+1}=Th_n$ converge to $e^{-cx}$ uniformly on compact subsets of $[0,+\infty)$.

The proof consists of several easy observations:

1. $\|Th\|_\infty\le\|h\|_\infty^2$ and, thereby, $\|h_n\|_\infty\le 1$ for all $n$.

2. If $h$ is bounded and $L$-Lipschitz, then $Th$ is $L\|h\|_\infty$-Lipschitz.

Indeed, $$ Th(s)-Th(S)=\frac1\pi\int_0^1\frac{dv}{\sqrt{v(1-v)}}[h(s(1-v))h(sv)-h(S(1-v))h(Sv)] $$ and $$ |h(s(1-v))h(sv)-h(S(1-v))h(Sv)| \\ \le\|h\|_\infty[|h(s(1-v))-h(S(1-v))|+|h(sv)-h(Sv)|] \\ \le\|h\|_\infty L[|s-S|(1-v+v)]=\|h\|_\infty L|s-S|\,. $$ Thus all $h_n$ are Lipschitz with the same Lipschitz constant $L$ as $h_0$.

3. If $A\in\mathbb R$ and $h(s)\ge e^{-As}$ on $[0,s_0]$, then $(Th)(s)\ge e^{-As}$ on $[0,s_0]$. 3') If $a\in\mathbb R$ and $0\le h(s)\le e^{-as}$ on $[0,s_0]$, then $(Th)(s)\le e^{-as}$ on $[0,s_0]$.

(all exponential functions are fixed points of $T$ and we have monotonicity as long as $h$ stays non-negative)

4. Now choose any $0<b<a<c<A<B$ and choose $s_0>0$ so that $e^{-As}\le h_0(s)\le e^{-as}$ on $[0,s_0]$ (by the derivative at zero condition such $s_0$ exists).

Consider the largest $S_n$ such that $e^{-Bs}\le h_n(s)\le e^{-bs}$ on $[0,S_n]$. We have $S_0\ge s_0$ and $S_{n+1}\ge S_n$. We want to improve the latter trivial inequality to some quantitative advance $S_{n+1}\ge S_n+\delta(S_n)$ where $\delta>0$ is separated from $0$ on any compact subinterval of $[s_0,+\infty)$. That is quite easy: $$ h_{n+1}(S_n)=(Th_n)(S_n)=\frac1\pi\int_0^{S_n}\frac{h_n(S_n-u)h_n(u)}{\sqrt{u(S_n-u)}}\,du\ge \\ \frac1\pi\int_0^{S_n}\frac{e^{-B(S_n-u)}e^{-Bu}}{\sqrt{u(S_n-u)}}\,du +\frac1\pi\int_0^{s_0}\frac{e^{-B(S_n-u)}[e^{-Au}-e^{-Bu}]}{\sqrt{u(S_n-u)}}\,du \\ \ge e^{-BS_n}+e^{-BS_n}S_n^{-1}\int_0^{s_0}[e^{-Au}-e^{-Bu}]\,du=e^{-BS_n}+\Delta(S_n) $$ Then, by the Lipschitz property of both $h_{n+1}$ and $e^{-Bs}$, the inequality $h_{n+1}(s)\ge e^{-Bs}$ persists on $[S_n,S_n+\delta(S_n)]$ with $\delta(S_n)=\frac{\Delta(S_n)}{B+L}$, say. The extension of the upper bound is similar.

The outcome is that the double inequality $e^{-Bs}\le h_n(s)\le e^{-bs}$ propagates from $[0,s_0]$ to the entire real line. Since $0<b<a<c<A<B$ were arbitrary, we conclude that $h_n$ tend to $e^{-cs}$ uniformly on compact intervals, finishing the story.