**A. Rosalsky, A. Volodin**

## On Convergence of Series of Random Elements via Maximal Moment Relations with Applications to Martingale Convergence and to Convergence of Series with -Orthogonal Summands

**abstract:**

The rate of convergence for an almost surely convergent series of
Banach space valued random elements is studied in this paper. As
special cases of the main result, known results are obtained for a
sequence of independent random elements in a Rademacher type $p$ Banach
space, and new results are obtained for a martingale difference sequence
of random elements in a martingale type $p$ Banach space and for a
$p$-orthogonal sequence of random elements in a Rademacher type $p$ Banach
space. The current work generalizes, simplifies, and unifies some of the
recent results of Nam and Rosalsky (1996) and Rosalsky and Rosenblatt
(1997, 1998).