On the a.s. Cesàro-\(\alpha\) convergence for stationary or orthogonal random variables
type: Item
instance of: scholarly article
title: On the a.s. Cesàro-\(\alpha\) convergence for stationary or orthogonal random variables
zbMATH Open document ID: 0684.60014
DOI: 10.1007/BF01051879
author: Yves Deniel
published in: Journal of Theoretical Probability
publication date: 1989-01-01T00:00:00Z
review text: Considered are a.s. Cesàro-\(\alpha\) convergences of stationary sequences \([X_ n]\) of random variables \(X_ n\) as \(n\to \infty\) and of stochastic processes X(t) as \(t\to \infty\) \((t>0)\). The random variables are given in the form \(X_ n=f\circ \partial^ n\) and the stochastic processes in the form \(f\circ \partial_ t\) with measurable measure- preserving transformations \(\partial\) and \(\partial_ t\) of a probability space (\(\Omega\),B,\(\mu)\). Further, f is a real function in \(L^ p\). Under the restriction \(0<\alpha \leq 1\), \(\alpha p<1\), local and universal ergodicity a.s. (and for \(L^ p\)-norm) in Cesàro-\(\alpha\) convergence are shown both for sequences and processes. For sequences, such convergence, in the case when the transformations were contractions, was earlier proved by R. Irmish. He then also showed by a counterexample that a.s. Cesàro-\(\alpha\) convergence failed in the case \(0<\alpha \leq 1\), \(\alpha p=1\). Here the author considers more general transformations and accordingly gives a more general counterexample.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 4121122
zbMATH Keywords: Cesàro-\(\alpha \) convergences of stationary sequences
zbMATH Keywords: measure- preserving transformations
zbMATH Keywords: counterexample
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4121122