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Monday, December 3, 2012

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}.




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