An $L^1$ ergodic theorem for sparse random subsequences

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 $ℓ^1$ convolution operators (Prop. 3.1).

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Published 1 January 2009