## Abstract

It is well-known that a probability measure μ on the circle T satisfies μn * f f dmp → 0 for every f ∈ Lp, every (some) p ∈ [1,∞), if and only if | μ(n)| < 1 for every non-zero n ∈ Z (μ is strictly aperiodic). In this paper we study the a.e. convergence of μn * f for every f ∈ Lp whenever p >1. We prove a necessary and sufficient condition, in terms of the Fourier- Stieltjes coefficients of μ, for the strong sweeping out property (existence of a Borel set B with lim supμn * 1B = 1 a.e. and lim infμn * 1B = 0 a.e.). The results are extended to general compact Abelian groups G with Haar measure m, and as a corollary we obtain the dichotomy: for μ strictly aperiodic, either μn * f → f dm a.e. for every p >1 and every f ∈ Lp(G,m), or μ has the strong sweeping out property.

Original language | English |
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Pages (from-to) | 550-568 |

Number of pages | 19 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 49 |

Issue number | 2 |

DOIs | |

State | Published - 1 May 2013 |

## Keywords

- Almost everywhere convergence
- Convolution powers
- Strictly aperiodic probabilities
- Sweeping out