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 $\alpha > 0$.
Publié le : 1983-08-14
Classification:
Orthogonal random variables,
Cesaro means of sequences,
strong law of large numbers,
60F15,
60G46
@article{1176993534,
author = {Moricz, Ferenc},
title = {On the Cesaro Means of Orthogonal Sequences of Random Variables},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 827-832},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993534}
}
Moricz, Ferenc. On the Cesaro Means of Orthogonal Sequences of Random Variables. Ann. Probab., Tome 11 (1983) no. 4, pp. 827-832. http://gdmltest.u-ga.fr/item/1176993534/