# ANNALES DE L'INSTITUT FOURIER

Almost Everywhere Convergence Of Convolution Powers Without Finite Second Moment
Annales de l'Institut Fourier, Volume 61 (2011) no. 2, p. 401-415

Bellow and Calderón proved that the sequence of convolution powers ${\mu }_{n}f\left(x\right)={\sum }_{k\in ℤ}{\mu }^{n}\left(k\right)f\left({T}^{k}x\right)$ converges a.e, when $\mu$ is a strictly aperiodic probability measure on $ℤ$ such that the expectation is zero, $E\left(\mu \right)=0$, and the second moment is finite, ${m}_{2}\left(\mu \right)<\infty$. In this paper we extend this result to cases where ${m}_{2}\left(\mu \right)=\infty$.

Nous généralisons un théorème de Bellow et Calderón concernant la convergence p.p. de puissances de convolution ${\mu }^{n}f\left(x\right)={\sum }_{k}{\mu }^{n}\left(k\right)f\left({T}^{k}x\right)$$T$ est une transformation préservant la mesure d’un espace de probabilités et $\mu$ est une mesure de probabilité sur les nombres entiers.

Accepted : 2010-01-05
DOI : https://doi.org/10.5802/aif.2618
Classification:  47A35
Keywords: Convolution powers, a.e convergence, Fourier transform, Lipschitz class Lip$\left(\alpha \right)$
@article{AIF_2011__61_2_401_0,
author = {Wedrychowicz, Christopher M.},
title = {Almost Everywhere Convergence  Of Convolution Powers Without Finite Second Moment},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {61},
number = {2},
year = {2011},
pages = {401-415},
doi = {10.5802/aif.2618},
mrnumber = {2895062},
zbl = {1242.47010},
language = {en},
url = {aif.centre-mersenne.org/item/AIF_2011__61_2_401_0}
}

Wedrychowicz, Christopher M. Almost Everywhere Convergence  Of Convolution Powers Without Finite Second Moment. Annales de l'Institut Fourier, Volume 61 (2011) no. 2, pp. 401-415. doi : 10.5802/aif.2618. https://aif.centre-mersenne.org/item/AIF_2011__61_2_401_0/

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