A weighted pointwise ergodic theorem
Let \(S\) be a measure-preserving transformation of a probability space \((Y,{\mathcal G},\nu)\). For \(g\in L^r(\nu)\), \(1<r<\infty\), the pointwise convergence of the weighted averages \[ {1\over N} \sum^N_{n= 1} X_ng(S^n y) \] is studied, where \((X_n)\) is a sequence of random variables on a probability space \(\Omega\). The main result asserts that if the \(X_n\)'s are i.i.d. and symmetric with \(E| X_1|^p< \infty\) for some \(p>1\), then there exists a subset of full measure \(\widetilde\Omega\subset\Omega\) such that for every dynamical system \((Y,{\mathcal G},\nu,S)\), for all \(1<r<\infty\), and for all \(g\in L^r(\nu)\) the convergence \[ {1\over N} \sum^N_{n= 1}X_n(\omega)g(S^ny)\to 0 \] holds \(\nu\)-a.e. whenever \(\omega\in\widetilde\Omega\). The proof relies on a maximal inequality of \textit{R. Salem} and \textit{A. Zygmund} [Acta Math. 91, 245-301 (1954; Zbl 0056.29001)] and on the following interesting property of i.i.d. sequences (Lemma 5): Let \(p>1\) and \(\beta\geq 2p/(p- 1)\) be an integer. If \((Y_n)\) is an i.i.d. sequence with \(E| Y_1|^p< \infty\) then the averages \({1\over N^{\beta-1}} \sum^{(N+ 1)^\beta}_{n= N^\beta} Y_n\) converge with probability 1. The main result remains an open problem in the case of \(p= 1\), \(r= 1\). It is shown however that in this situation the \(L^1(\nu)\)-convergence does hold even if the i.i.d. sequence \(X_n\) is replaced by any stationary ergodic sequence with finite expectation.
scientific article; zbMATH DE number 1156747
scientific article