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

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