The Lévy-Baxter theorem and decomposition of Gaussian processes. (English. Russian original) Zbl 0625.60009

Ryzhov, Yu. M.

Theory Probab. Math. Stat. 34, 139-145 (1987); translation from Teor. Veroyatn. Mat. Stat. 34, 122-129 (1986).

The paper gives asymptotic expansions for probabilities that sums of independent random elements taking values in a separable Hilbert space H belong to different subsets of H.
Let \(X_ 1,X_ 2,...,X_ n\) be independent random elements of H and \(Y_ 1,Y_ 2,...,Y_ n\) independent Gaussian random elements of H with zero means and covariance operators \(A_ i\), \(i=1,2,...,n\). Put \(S_ n=\sum^{n}_{i=1}X_ i\), \(Z_ n=\sum^{n}_{i=1}Y_ i.\)
Let \(D: H\to H\) be a linear bounded symmetric operator with the norm \(| D| =\sup_{| x| =1}| (Dx,x)|\); \(d: H\to R\) a linear continuous functional. Put \(W(x)=(Dx,x)+d(x)\), \(K_{a,r}=\{x\in H:\) \(W(x+a)<r\}\), \(r\geq 0\). \(a\in H\). There are given conditions under which for \(n\geq 1\), \(a\in H\), \(r\in {\mathbb{R}}:\)
(1) \(| P(S_ n\in K_{a,r})-P(Z_ n\in K_{a,r})| \leq C(1+\| a\|^ 3)(\sum^{n}_{i=1}E\| X_ i\|^ 2 I[\| X_ i\| \geq 1]+\sum^{n}_{i=1}E\| X_ i\|^ 3 I[\| X_ i\| <1]);\text{ where } C\quad is\quad positive\quad cons\tan t,\) \[ (2)\quad P[S_ n\in K_{a,r}] = P[Z_ n\in K_{a,r}]+b+\Delta, \]
\[ where\quad | b| \quad <\quad C_ 1(1+\| a\|^ 3)\sum^{n}_{i=1}E\| X_ i\|^ 3 I[\| X_ i\| <1], \]
\[ | \Delta | \quad <\quad C_ 2((1+\| a\|^ 3))(\sum^{n}_{i=1}E\| X_ i\|^ 3 I[\| X_ i\| <1])^{2-\epsilon}\quad + \]
\[ +\quad (1+\| a\|^ 6)(\sum^{n}_{i=1}E\| X_ i\|^ 2 I[\| X_ i\| \geq 1]\quad +\sum^{n}_{i=1}E\| X_ i\|^ 4 I[\| X_ i\| \geq 1], \] \(C_ 1\) and \(C_ 2\) are positive constants;
\[ (3)\quad P[S_ n\in K_{a,r}] = P[Z_ n\in K_{a,r}]\quad +\sum^{m-2}_{i=1}b_ i+\Delta, \] where \(b_ i\) and \(\Delta\) are precisely defined in the paper.

Reviewer: D.Szynal

MSC:

60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
60F25	\(L^p\)-limit theorems
60G15	Gaussian processes

Keywords:

asymptotic expansions; Gaussian random elements; covariance operators