Degeneration of Gaussian measures

$\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\ve}{{\varepsilon}}$ $\newcommand{\bsV}{\boldsymbol{V}}$  Suppose that $\bsV$ is an $N$-dimensional real Euclidean space   equipped with an orthogonal  direct sum $\newcommand{\bsU}{\boldsymbol{U}}$ $\newcommand{\bsW}{\boldsymbol{W}}$

$$\bsV =\bsU\oplus \bsW. \tag{1}\label{1}$$

Suppose that $ S_n: \bsU\to\bsU $ and $C_n:\bsW\to \bsW$ are symmetric positive definite  operators such that

$$S_n\to 0,\;\;C_n\to C,\;\;\mbox{as}\;\;n\to \infty $$

where $C$ is a symmetric positive definite operator on $\bsW$.     We set


$$A_n=S_n\oplus C_n :\bsV\to \bsV $$

and we think of $A_n$ as the covariance   matrix  of a  Gaussian measure on $\bsV$ $\newcommand{\bv}{\boldsymbol{v}}$

$$\gamma_{A_n}(|d\bv|)=\frac{1}{\sqrt{\det 2\pi A_n}}  e^{-\frac{1}{2}(A_n^{-1} \bv,\bv)} |d\bv|. $$

Suppose that $f:\bsV\to \bR$ is a  locally Lipschitz function,   positively homogeneous of degree $k\geq 1$.

 I am interested in the  behavior as $n\to \infty$ of the expectation

$$ E_n(f):=\int_{\bsV} f(\bv)\gamma_{A_n}(|d\bv|). $$

$\newcommand{\bu}{\boldsymbol{u}}$  $\newcommand{\bw}{\boldsymbol{w}}$ We respect to the decomposition (\ref{1}) a vector  $\bv\in \bv_0$  can be written as an orthogonal sum $\bv=\bu+\bw$.

Define

$$\bar{f}_n:\bsW\to [0,\infty),\;\; \bar{f}_n(\bw)= \int_{\bsU} f(\bu+\bw) \gamma_{S_n}(|d\bu|),  $$

where $d\gamma_{S_n}$ denotes the Gaussian measure  on $\bsU$ with  covariance form $S_n$.  Then


$$ E_n(f)=\int_{\bsW} \bar{f}_n(\bw) \gamma_{C_n}(|d\bw|). \tag{2}\label{2}  $$

For $\bw\in \bsW$ and $r\in (0,1]$ we set

$$m(\bw, r) := \sup_{|\bu|\leq r}|f(\bw+u)- f(u)|.  $$

Note that

$$ \exists L>0: m(\bw,r)\leq   Lr,\;\;\forall |\bw|= 1\tag{3}\label{3} $$

In general,   we set $\bar{\bw}:=\frac{1}{|\bw|} \bw$. If $|\bu|\leq r$  and we have
$$
\bigl|\;f(\bw+\bu)-g(\bw) \;\bigr|= |\bw|^k \left| f\Bigl(\bar{\bw}+\frac{1}{|\bw|} \bu\Bigr) -f(\bar{\bw})\right| \leq  L |\bw|^{k-1} r,
$$
so that
$$
m(w,r) \leq L|\bw|^{k-1} r,\;\;\forall \bw\in\bsW,\;\;r\in (0,1].  \tag{4}\label{4}
$$

To proceed further, we need a vector counterpart for the Chebysev inequality.

Lemma 1.  Suppose $S:\bsU\to \bsU$ is a  symmetric, positive definite operator. We set $R:=S^{-\frac{1}{2}}$ and  denote by $\gamma_{S}$ the associated   Gaussian measure. Then for any $c,\ell>0$  we have $\newcommand{\bsi}{\boldsymbol{\sigma}}$

$$ \int_{ |R \bu|\geq c}  |\bu|^\ell d\gamma_S(|\bu|) \leq \sqrt{2^{\ell+m-\frac{3}{2}} \Gamma\Bigl(\; \ell+m-\frac{1}{2}\;\Bigr)} \frac{\bsi_{m-1}}{(2\pi)^{\frac{m}{2}}} \Vert S\Vert^{\frac{\ell}{2}}c^{-\frac{1}{2}}e^{-\frac{c^2}{4}} , \tag{5}\label{5}$$

where $m=\dim\bsU$ and  and $\bsi_N$ denote the area of the $N$-dimensional unit sphere.

Proof.   We make the change in variables $\newcommand{\bx}{\boldsymbol{x}}$  $\bx:=R\bu$ and we  deduce
$$
\int_{ |R \bu|\geq c}  |\bu|^\ell d\gamma_S(|\bu|)\leq \frac{1}{(2\pi)^{\frac{m}{2}}} \int_{|\bx|\geq c} |S^{\frac{1}{2}} \bx|^\ell  e^{-\frac{1}{2}|\bx|^2} |d\bx| $$

$$
\leq \frac{\Vert|S\Vert^{\frac{\ell}{2}}}{(2\pi)^{\frac{m}{2}}} \int_{|\bx|\geq c} |\bx|^\ell  e^{-\frac{1}{2}|\bx|^2} |d\bx|=\frac{\bsi_{m-1}\Vert S\Vert^{\frac{\ell}{2}}}{(2\pi)^{\frac{m}{2}}}\int_{t>c} t^{\ell+m-1} e^{-\frac{1}{2} t^2} dt
$$
$$
\leq  \frac{\bsi_{m-1}\Vert S\Vert^{\frac{\ell}{2}}}{(2\pi)^{\frac{m}{2}}} \left(\int_{t>c}  e^{-\frac{1}{2} t^2} dt\right)^{\frac{1}{2}}\left(\int_{t>0} t^{2\ell+2m-2} e^{-\frac{1}{2} t^2} dt\right)^{\frac{1}{2}}
$$
Now observe that we have
$$
\int_{t>c}  e^{-\frac{1}{2} t^2} dt \leq \frac{1}{c} e^{-\frac{c^2}{2}},
$$
and  using the change of variables $s=\frac{t^2}{2}$ we  deduce

$$ \int_{t>0} t^{2\ell+2m-2} e^{-\frac{1}{2} t^2} dt =2^{\ell+m-\frac{3}{2}}\int_0^\infty s^{\ell+m-\frac{1}{2}-1} e^{-s} ds= 2^{\ell+m-\frac{3}{2}} \Gamma( \ell+m-\frac{1}{2}). $$

This proves the lemma. q.e.d




We now want to compare $\bar{f}_n(\bw)$ and $f(\bw)$ for $\bw\in\bsW$.  We plan to use Lemma  1.   Set $R_n:=S_n^{-\frac{1}{2}}$ and $m:=\dim\bsU$.   Observe that

$$|\bu\|= |S_n^{\frac{1}{2}}R_n\bu|\leq \Vert S_n^{\frac{1}{2}}\Vert\cdot |R_n\bu|. $$

For simplicity   set $s_n:=  \Vert S_n^{\frac{1}{2}}\Vert$.   Choose a sequence of positive numbers  $c_n$ such that $c_n\to\infty$ and  $ s_n c_n\to 0$.  Later  we will add several requirements to this sequence.

$$
\bigl|\;\bar{f}_n(\bw)-f(\bw)\;\bigr|=\left| \int_{\bsU} (\; f(\bw+\bu)- f(\bw)\; ) \gamma_{S_n}(|d\bu|)\right| $$
$$
\leq \left| \int_{|R_n\bu|\leq c_n}  (\; f(\bw+\bu)- f(\bw)\; ) \gamma_{S_n}(|d\bu|)\right|+\left|\int_{|R_n\bu|\geq c_n} (\; f(\bw+\bu)- f(\bw)\; ) \gamma_{S_n}(|d\bu|)\right| $$
$$\stackrel{(\ref{4})}{\leq} L|\bw|^{k-1}s_n c_n +C \int_{|R_n\bu|\geq c_n}(|\bw|^k+|\bu|^k) \gamma_{S_n}(|d\bu|) $$

$$
\stackrel{(\ref{5})}{\leq}  L|\bw|^{k-1}s_n c_n + Z(k, m)c_n^{-\frac{1}{2}} e^{-\frac{c_n^2}{4}}(1+s_n^k),
$$
where $Z(k,m)$ is a constant that depends only on   $k$ and $m$.


We deduce that there exists a constant $C>0$ independent of   $n,w$  such that  for any sequence $c_n\to \infty$ such that $s_nc_n\to 0$, $s_n:=\Vert S_n\Vert^{\frac{1}{2}}$ we have

$$
\bigl|\;\bar{f}_n(\bw)-f(\bw)\;\bigr| \leq C\bigl(\; |\bw|^{k-1}s_nc_n + e^{-\frac{c_n^2}{4}}\;\bigr). \tag{6}\label{6}

$$
We deduce that

$$

\Bigl|\; E_n(f) -\int_{\bsW} f(\bw) \gamma_{C_n}(|d\bw|)\;\Bigr|  \leq   C\left(s_nc_n\int_{\bsW} |\bw|^{k-1} \gamma_{C_n}(|d\bw|) + e^{-\frac{c_n^2}{4}}\;\right).\tag{7}\label{7}
$$

Finally let us estimate

$$D_n:=\int_{\bsW} f(\bw) \gamma_{C_n}(|d\bw|)-\int_{\bsW} f(\bw) d\gamma_{C}(|d\bw|). $$

We have  $\newcommand{\one}{\boldsymbol{1}}$
$$D_n= \int_{\bsW} \left( f\bigl( C_n^{\frac{1}{2}}\bw\;\bigr)-f\bigl( C^{\frac{1}{2}}\bw\;\bigr) \;\right)\gamma_{\one}(|\bw|)
$$
and we conclude that
$$
\left|\; \int_{\bsW} f(\bw) d\gamma_{C_n}(|d\bw|)-\int_{\bsW} f(\bw) d\gamma_{C}(|d\bw|)\;\right| \leq  L \Bigl\Vert \;C_n^{\frac{1}{2}}-C^\frac{1}{2}\;\Bigr\Vert \int_{\bsW}|\bw|^k \gamma_{\one}(|d\bw|). \tag{8}\label{8}
$$

In (\ref{7}) we let $c_n:=s_n^{-\ve}$. If we denote by $A_\infty$ the limit of the covariance matrices $A_n$, $A=\lim_{n\to\infty} A_n =0\oplus C$, then we deduce from the above computations that for any $\ve>0$ there exists a  constant $C_\ve>0$ such that
$$
\left|\; \int_{\bsV} f(\bv) \gamma_{A_n} (|d\bw|) -\int_{\bsV} f(\bv) \gamma_{A_\infty} (|d\bw|) \;\right|\leq C_\ve \left(s_n^{1-\ve}+ \Bigl\Vert \;C_n^{\frac{1}{2}}-C^\frac{1}{2}\;\Bigr\Vert\right)\leq C_\ve \Bigl\Vert A_n^{\frac{1}{2}}-A_\infty^{\frac{1}{2}}\Bigr\Vert^{1-\ve}.\tag{9}\label{9}
$$
This can be generalized a bit. Suppose that $T_n:\bsU\to \bsU$ is a sequence of orthogonal operators such that  $T_n\to \one_{\bsU}$

Using (\ref{7})  we deduce

$$ \left|\;\int_{\bsV} T^*_nf(\bv) \gamma_{A_n}(|d\bv|)-\int_{\bsV}  f(\bv) \gamma_{A_n}(|d\bv|)\right|= \left| \int_{\bsV} f(T_n A_n^{\frac{1}{2}}\bx)-  f( A_n^{\frac{1}{2}}\bx)\gamma_{\one}(|d\bx|) \right| \leq L \Bigl\Vert A_n^{\frac{1}{2}}\Bigr\Vert \Vert T_n-\one\Vert.
$$
Observe that
$$
\int_{\bsV} T^*_nf(\bv) \gamma_{A_n}(|d\bv|)=\int_{\bsV} f(\bv) \gamma_{B_n}(|d\bv|),
$$
where
$$
B_n= T_nA_nT_n^*.
$$

Suppose that we are in the fortunate case when $f|_{\bsW}=0$.    Then

$$\int_{\bsW} f(\bw) d\gamma_{C_n}(|d\bw|)=\int_{\bsW} f(\bw) d\gamma_{C}(|d\bw|)=0 $$

and (\ref{9})  can be improved to

$$
\left|\; \int_{\bsV} f(\bv) \gamma_{A_n} (|d\bw|)\right|\leq C_\ve s_n^{1-\ve}.
$$