On the a.s. Cesàro-\(\alpha\) convergence for stationary or orthogonal random variables
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.
Cesàro-\(\alpha \) convergences of stationary sequences
scientific article; zbMATH DE number 4121122