Divergence Criterion for a Class of Random Series Related to the Partial Sums of I.I.D. Random Variables

Abstract

Let $\{X, X_{n};~n \ge 1 \}$ be a sequence of independent and identically distributed Banach space valued random variables. This paper is devoted to providing a divergence criterion for a class of random series of the form $\sum _{n=1}^{\infty } f_{n}\left( \left\| S_{n} \right\| \right)$ where $S_{n} = X_{1} + \cdots + X_{n}, ~n \ge 1$ and $\left\{ f_{n}(\cdot ); n \ge 1 \right\}$ is a sequence of nonnegative nondecreasing functions defined on $[0, \infty )$. More specifically, it is shown that (i) the above random series diverges almost surely if $\sum _{n=1}^{\infty } f_{n} \left( cn^{1/2} \right) = \infty$ for some $c > 0$ and (ii) the above random series converges almost surely if $\sum _{n=1}^{\infty } f_{n} \left( cn^{1/2} \right) < \infty$ for some $c > 0$ provided additional conditions are imposed involving X, the sequences $\left\{ S_{n};~n \ge 1 \right\}$ and $\left\{ f_{n}(\cdot ); n \ge 1 \right\}$, and c. A special case of this criterion is a divergence/convergence criterion for the random series $\sum _{n=1}^{\infty } a_{n} \left\| S_{n} \right\| ^{q}$ based on the series $\sum _{n=1}^{\infty } a_{n} n^{q/2}$ where $\left\{ a_{n};~n \ge 1 \right\}$ is a sequence of nonnegative numbers and $q > 0$.