The Annals of Probability

On the Cesaro Means of Orthogonal Sequences of Random Variables

Ferenc Moricz

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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 $\alpha > 0$.

Article information

Source
Ann. Probab., Volume 11, Number 3 (1983), 827-832.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993534

Digital Object Identifier
doi:10.1214/aop/1176993534

Mathematical Reviews number (MathSciNet)
MR704576

Zentralblatt MATH identifier
0514.60034

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G46: Martingales and classical analysis

Keywords
Orthogonal random variables Cesaro means of sequences strong law of large numbers

Citation

Moricz, Ferenc. On the Cesaro Means of Orthogonal Sequences of Random Variables. Ann. Probab. 11 (1983), no. 3, 827--832. doi:10.1214/aop/1176993534. https://projecteuclid.org/euclid.aop/1176993534


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