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 μ n f(x)= k μ n (k)f(T k x) converges a.e, when μ is a strictly aperiodic probability measure on such that the expectation is zero, E(μ)=0, and the second moment is finite, m 2 (μ)<. In this paper we extend this result to cases where m 2 (μ)=.

Nous généralisons un théorème de Bellow et Calderón concernant la convergence p.p. de puissances de convolution μ n f(x)= k μ n (k)f(T k x)T est une transformation préservant la mesure d’un espace de probabilités et μ est une mesure de probabilité sur les nombres entiers.

Received : 2009-07-01
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(α)
@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/

[1] Bellow, Alexandra; Calderón, Alberto P. A weak-type inequality for convolution products, Harmonic analysis and partial differential equations (Chicago, IL, 1996) (Chicago Lectures in Math.), Univ. Chicago Press, Chicago, IL, 1999, pp. 41-48 | MR 1743854 | Zbl 0960.28011

[2] Bellow, Alexandra; Jones, Roger L.; Rosenblatt, Joseph Almost everywhere convergence of weighted averages, Math. Ann., Tome 293 (1992) no. 3, pp. 399-426 | Article | MR 1170516 | Zbl 0736.28008

[3] Bellow, Alexandra; Jones, Roger L.; Rosenblatt, Joseph Almost everywhere convergence of convolution powers, Ergodic Theory Dynam. Systems, Tome 14 (1994) no. 3, pp. 415-432 | Article | MR 1293401 | Zbl 0818.28005

[4] Foguel, Shaul R. On iterates of convolutions, Proc. Amer. Math. Soc., Tome 47 (1975), pp. 368-370 | Article | MR 374816 | Zbl 0299.43004

[5] Losert, V. A remark on almost everywhere convergence of convolution powers, Illinnois J. Math., Tome 43 (1999) no. 3, pp. 465-479 | MR 1700602 | Zbl 0963.28014

[6] Losert, V. The strong sweeping out property for convolution powers, Ergodic Theory Dynam. Systems, Tome 21 (2001) no. 1, pp. 115-119 | Article | MR 1826663 | Zbl 0972.37002

[7] Móricz, Ferenc Absolutely convergent Fourier series and function classes, J. Math. Anal. Appl., Tome 324 (2006) no. 2, pp. 1168-1177 | Article | MR 2266550 | Zbl 1103.42003

[8] Petrov, V. V. Sums of independent random variables, Springer-Verlag, New York, 1975 (Translated from the Russian by A. A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82) | MR 388499 | Zbl 0322.60042

[9] Young, W.H. On Indeterminate Forms, Proc. London Math. Soc., Tome s2-8(1) (1910), pp. 40-76 | Article

[10] Zygmund, A. Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959 | MR 107776 | Zbl 0085.05601