On convergence of series of dependent random elements in Hilbert spaces. (English. Russian original) Zbl 0635.60027

Klesov, O. I.

Theory Probab. Math. Stat. 32, 35-43 (1986); translation from Teor. Veroyatn. Mat. Stat. 32, 33-42 (1985).

A sequence \(\{X_ n\}\) taking values in a Hilbert space \(\Gamma\) is called quasistationary if E \(X_ n=0\), and \(\sup_ n E(X_ nX_{n+m})\leq \phi (m)\) for \(m\geq 0\). The author gives some sufficient conditions for a series \(\sum X_ n\) of quasistationary random elements in \(\Gamma\) to converge almost surely in the norm of \(\Gamma\). He also considers the rate of convergence and divergence of the series. The proof is based on an inequality which can be regarded as an analogue of Men’shov-Rademacher’s inequality.

Reviewer: T.Mori

MSC:

60F15	Strong limit theorems
60G10	Stationary stochastic processes
60G50	Sums of independent random variables; random walks

Keywords:

series of random vectors; random elements in Hilbert space; quasistationary sequence; orthogonal sequence; rate of convergence; Men’shov-Rademacher’s inequality