On the Cesaro Means of Orthogonal Sequences of Random Variables

Abstract

Let $\{\xi_k: k \geq 0\}$ be an orthogonal sequence of random variables with finite second moments $E\xi^2_k = \sigma^2_k$. It is well-known that if $\sum^\infty_{k=0} \sigma^2_k(k + 1)^{-2}\lbrack\log(k + 2)\rbrack^2 < \infty$, then the first arithmetic means $\tau^0_n: = (n + 1)^{-1} \sum^n_{k=0} \xi_k \rightarrow 0$ a.s. $(n \rightarrow \infty)$. Now we prove that the means $\tau^1_n: = (n + 1)^{-1} \sum^n_{k=0} (1 - k(n + 1)^{-1})\xi_k \rightarrow 0$ a.s. $(n \rightarrow \infty)$ merely under the condition $\sum^\infty_{k=0} \sigma^2_k(k + 1)^{-2} < \infty$. We define the means $\tau^\alpha_n$ for every real $\alpha$, too and prove that under the latter condition $\tau^\alpha_n \rightarrow 0$ a.s. $(n \rightarrow \infty)$ provided $\alpha > 0$.