An L^1 Ergodic Theorem for Sparse Random Subsequences

Abstract

We prove an L^1 subsequence ergodic theorem for sequences chosen by independent random selector variables, thereby showing the existence of universally L^1-good sequences nearly as sparse as the set of squares. In the process, we prove that a certain deterministic condition implies a weak maximal inequality for a sequence of \ell^1 convolution operators.

LaTeX, 9 pages