## A weighted pointwise ergodic theorem.(English)Zbl 0902.28011

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 R. Salem and 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.

### MSC:

 28D05 Measure-preserving transformations 60G50 Sums of independent random variables; random walks

Zbl 0056.29001
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