Almost Everywhere Convergence Of Convolution Powers Without Finite Second Moment
[La convergence presque partout de pouvoirs de convolution sans deuxième moment fini]
Wedrychowicz, Christopher M.1
1 Indiana University South Bend Department of Mathematical Sciences 1700 Mishawaka Ave. South Bend 46634 (USA)
Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 401-415.

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) où T est une transformation préservant la mesure d’un espace de probabilités et μ est une mesure de probabilité sur les nombres entiers.

Bellow and Calderón proved that the sequence of convolution powers μ n f(x)=∑ k∈ℤ μ n (k)f(T k x) converges a.e, when μ is a strictly aperiodic probability measure on ℤ 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)=∞.

DOI : 10.5802/aif.2618
Classification : 47A35
Keywords: Convolution powers, a.e convergence, Fourier transform, Lipschitz class Lip$(\alpha )$
Mot clés : pouvoirs de convulions, convergence p.p, transformée de Fourier, la classe de Lipschitz Lip$(\alpha )$
Wedrychowicz, Christopher M. 1

1 Indiana University South Bend Department of Mathematical Sciences 1700 Mishawaka Ave. South Bend 46634 (USA) 
@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},
     pages = {401--415},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {2},
     year = {2011},
     doi = {10.5802/aif.2618},
     zbl = {1242.47010},
     mrnumber = {2895062},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2618/}
}
TY  - JOUR
AU  - Wedrychowicz, Christopher M.
TI  - Almost Everywhere Convergence  Of Convolution Powers Without Finite Second Moment
JO  - Annales de l'Institut Fourier
PY  - 2011
SP  - 401
EP  - 415
VL  - 61
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2618/
DO  - 10.5802/aif.2618
LA  - en
ID  - AIF_2011__61_2_401_0
ER  - 
%0 Journal Article
%A Wedrychowicz, Christopher M.
%T Almost Everywhere Convergence  Of Convolution Powers Without Finite Second Moment
%J Annales de l'Institut Fourier
%D 2011
%P 401-415
%V 61
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2618/
%R 10.5802/aif.2618
%G en
%F AIF_2011__61_2_401_0
Wedrychowicz, Christopher M. Almost Everywhere Convergence  Of Convolution Powers Without Finite Second Moment. Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 401-415. doi : 10.5802/aif.2618. http://www.numdam.org/articles/10.5802/aif.2618/

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

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

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

Cité par Sources :