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# Mathematical Research Letters

## Volume 16 (2009)

### Number 5

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

Pages: 849 – 859

DOI: http://dx.doi.org/10.4310/MRL.2009.v16.n5.a8

#### Author

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