Almost everywhere convergence of convolution powers on compact abelian groups
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 550-568.

Il est connu qu’une mesure de probabilité μ sur le cercle 𝕋 satisfait μ n *f-fdm p 0 pour toute fonction fL p et pour tout p[1,) (ou pour un p[1,)), si et seulement si μ est strictement apériodique (i.e. |μ ^(n)|<1 pour tout n non nul dans ). Nous étudions ici la convergence presque partout de μ n *f pour fL p , p>1. Nous montrons une condition nécessaire et suffisante portant sur les coefficients de Fourier-Stieltjes de μ pour la propriété de “balayage fort” (existence d’un borélien B tel que lim supμ n *1 B =1 p.p. et lim infμ n *1 B =0 p.p.). Les résultats sont étendus aux groupes abéliens compacts généraux G de mesure de Haar m. Comme corollaire nous obtenons la dichotomie suivante : pour μ strictement apériodique, soit μ n *ffdm p.p. pour tout p>1 et toute fonction fL p (G,m), soit μ vérifie la propriété de balayage fort.

It is well-known that a probability measure μ on the circle 𝕋 satisfies μ n *f-fdm p 0 for every fL p , every (some) p[1,), if and only if |μ ^(n)|<1 for every non-zero n (μ is strictly aperiodic). In this paper we study the a.e. convergence of μ n *f for every fL p 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 supμ n *1 B =1 a.e. and lim infμ n *1 B =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 μ n *ffdm a.e. for every p>1 and every fL p (G,m), or μ has the strong sweeping out property.

DOI : 10.1214/11-AIHP468
Classification : 37A30, 28D05, 47A35, 60G50, 42A38
Mots clés : convolution powers, almost everywhere convergence, sweeping out, strictly aperiodic probabilities
@article{AIHPB_2013__49_2_550_0,
     author = {Conze, Jean-Pierre and Lin, Michael},
     title = {Almost everywhere convergence of convolution powers on compact abelian groups},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {550--568},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {2},
     year = {2013},
     doi = {10.1214/11-AIHP468},
     mrnumber = {3088381},
     zbl = {1281.37005},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/11-AIHP468/}
}
TY  - JOUR
AU  - Conze, Jean-Pierre
AU  - Lin, Michael
TI  - Almost everywhere convergence of convolution powers on compact abelian groups
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2013
SP  - 550
EP  - 568
VL  - 49
IS  - 2
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/11-AIHP468/
DO  - 10.1214/11-AIHP468
LA  - en
ID  - AIHPB_2013__49_2_550_0
ER  - 
%0 Journal Article
%A Conze, Jean-Pierre
%A Lin, Michael
%T Almost everywhere convergence of convolution powers on compact abelian groups
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2013
%P 550-568
%V 49
%N 2
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/11-AIHP468/
%R 10.1214/11-AIHP468
%G en
%F AIHPB_2013__49_2_550_0
Conze, Jean-Pierre; Lin, Michael. Almost everywhere convergence of convolution powers on compact abelian groups. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 550-568. doi : 10.1214/11-AIHP468. http://archive.numdam.org/articles/10.1214/11-AIHP468/

[1] M. Anoussis and D. Gatzouras. A spectral radius formula for the Fourier transform on compact groups and applications to random walks. Adv. Math. 188 (2004) 425-443. | MR | Zbl

[2] R. N. Bhattacharya. Speed of convergence of the n-fold convolution of a probability measure on a compact group. Z. Wahrsch. Verw. Gebiete 25 (1972) 1-10. | MR | Zbl

[3] A. Bellow and R. Jones. A Banach principle for L . Adv. Math. 120 (1996) 155-172. | MR | Zbl

[4] A. Bellow, R. Jones and J. Rosenblatt. Almost everywhere convergence of powers. In Almost Everywhere Convergence 99-120. G. Edgar and L. Sucheston (Eds). Academic Press, Boston, MA, 1989. | MR | Zbl

[5] A. Bellow, R. Jones and J. Rosenblatt. Almost everywhere convergence of convolution powers. Ergodic Theory Dynam. Systems 14 (1994) 415-432. | MR | Zbl

[6] D. Burkholder. Successive conditional expectations of an integrable function. Ann. Math. Statist. 33 (1962) 887-893. | MR | Zbl

[7] G. Cohen. Iterates of a product of conditional expectation operators. J. Funct. Anal. 242 (2007) 658-668. | MR | Zbl

[8] A. Del Junco and J. Rosenblatt. Counterexamples in ergodic theory and number theory. Math. Ann. 245 (1979) 185-197. | MR | Zbl

[9] Y. Derriennic and M. Lin. Variance bounding Markov chains, L 2 -uniform mean ergodicity and the CLT. Stoch. Dyn. 11 (2011) 81-94. | MR | Zbl

[10] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers, 5th edition. Clarendon Press, New York, 1979. | JFM | MR | Zbl

[11] S. Horowitz. Pointwise convergence of the iterates of a Harris-recurrent Markov operator. Israel J. Math. 33 (1979) 177-180. | MR | Zbl

[12] R. Jones, J. Rosenblatt and A. Tempelman. Ergodic theorems for convolutions of a measure on a group. Illinois J. Math. 38 (1994) 521-553. | MR | Zbl

[13] U. Krengel. Ergodic Theorems. De Gruyter, Berlin, 1985. | MR | Zbl

[14] M. Lin. The uniform zero-two law for positive operators in Banach lattices. Studia Math. 131 (1998) 149-153. | MR | Zbl

[15] V. Losert. A remark on almost everywhere convergence of convolution powers. Illinois J. Math. 43 (1999) 465-479. | MR | Zbl

[16] V. Losert. The strong sweeping out property for convolution powers. Ergodic Theory Dynam. Systems 21 (2001) 115-119. | MR | Zbl

[17] D. Ornstein. On the pointwise behavior of iterates of a self-adjoint operator. J. Math. Mech. 18 (1968) 473-477. | MR | Zbl

[18] V. Oseledets. Markov chains, skew products and ergodic theorems for “general” dynamic systems. Theory Probab. Appl. 10 (1965) 499-504. | MR | Zbl

[19] F. Parreau. Measures with real spectra. Invent. Math. 98 (1989) 311-330. | MR | Zbl

[20] J. Rosenblatt. Ergodic group actions. Arch. Math. (Basel) 47 (1986) 263-269. | MR | Zbl

[21] M. Rosenblatt. Markov Processes. Structure and Asymptotic Behavior. Springer, Berlin, 1971. | MR | Zbl

[22] K. A. Ross and D. Xu. Norm convergence of random walks on compact hypergroups. Math. Z. 214 (1993) 415-423. | MR | Zbl

[23] J.-C. Rota. An “Alternierende Verfahren” for general positive operators. Bull. Amer. Math. Soc. 68 (1962) 95-102. | MR | Zbl

[24] W. Rudin. Fourier Analysis on Groups. Interscience, New York, 1962. | MR | Zbl

[25] E. M. Stein. On the maximal ergodic theorem. Proc. Natl. Acad. Sci. USA 47 (1961) 1894-1897. | MR | Zbl

[26] N. Varopoulos. Sets of multiplicity in locally compact Abelian groups. Ann. Inst. Fourier 16 (1966) 123-158. | Numdam | MR | Zbl

Cité par Sources :