Series with independent summands in Hilbert spaces. (Russian) Zbl 0577.60008

Buldygin, V. V.

Teor. Veroyatn. Mat. Stat. 29, 14-27 (1983).

Let \(\{X_ k\}\) be a sequence of independent random variables in a Hilbert space H. Let \(\phi_ k\) be the characteristic measure of the distribution \(\mu_ k\) of \(X_ k.\)
The main theorem of this work states: For the almost sure convergence of \(\sum_{k=1}X_ k\) it is necessary that for an arbitrary positively determined selfadjoint kernel operator S from H to H and sufficient that for all such operators the infinite product \(\prod_{k=1}Y_ k\) converges, where \(Y_ k\) are random variables derived from \(\phi_ k\) and S, on a certain, on S depending set.
This result is applied to series with symmetrical summands and to series with stable summands.

Reviewer: C.Baldauf

MSC:

60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15	Strong limit theorems

Keywords:

independent random variables in a Hilbert space; almost sure convergence; stable summands