Israel Journal of Mathematics

, Volume 101, Issue 1, pp 157–178 | Cite as

Theoremes ergodiques perturbes

Article

Abstract

We obtain results on almost sure convergence of ergodic averages along arithmetic subsequences perturbed by independent identically distributed random variables having ap th finite moment for somep>0. To prove these results, we use methods based on the harmonic analysis and the theory of Gaussian processes. In fact that will express the stability of Bourgain’s results concerning convergence of ergodic averages for certain arithmetic subsequences.

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References

  1. [B1] J. Bourgain,Pointwise ergodic theorems for arithmetic sets with an appendix on return time sequences, Publications Mathématiques de l’Institut des Hautes Études Scientifiques69 (1988), 1–48.Google Scholar
  2. [B2] J. Bourgain,An approach to pointwise ergodic theorems, Lecture Notes in Mathematics1317, Springer-Verlag, Berlin, 1988, pp. 204–223.Google Scholar
  3. [B3] J. Bourgain,On the pointwise ergodic theorem on L p for arithmetic sets, Israel journal of Mathematics61 (1988), 73–84.MATHMathSciNetGoogle Scholar
  4. [B4] J. Bourgain,Almost sure convergence and bounded entropy, Israel Journal of Mathematics63 (1988), 79–87.MATHMathSciNetGoogle Scholar
  5. [F1] X. Fernique,Régularité de fonctions aléatoires gaussiennes stationnaires, Probability Theory and Related Fields88 (1991), 521–536.MATHCrossRefMathSciNetGoogle Scholar
  6. [F2] X. Fernique,Un exemple illustrant l’emploi des méthodes gaussiennes, à paraître dans les actes de la “Conférence en l’honneur de J. P. Kahane”, Paris, Juillet, 1993.Google Scholar
  7. [F3] X. Fernique,Une majoration des fonctions aléatoires gaussiennes à valeurs vectorielles, Comptes Rendus de l’Académie des Sciences, Paris300 (1985), 315–318.MATHMathSciNetGoogle Scholar
  8. [F4] X. Fernique,Régularité de fonctions aléatoires gaussiennes à valeurs vectorielles, inProbability Theory on Vector Spaces IV, Lectures Notes in Mathematics1391, Springer-Verlag, Berlin, 1989, pp. 63–73.CrossRefGoogle Scholar
  9. [F5] X. Fernique,Régularité des trajectoires de fonctions aléatoires gaussiennes, Lecture Notes in Mathematics480, Springer-Verlag, Berlin, 1974, pp. 1–97.Google Scholar
  10. [K] U. Krengel,Ergodic Theorems, W. de Gruyter, 1985.Google Scholar
  11. [L] E. Lesigne,Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory and Dynamical Systems13 (1993), 767–784.MATHMathSciNetGoogle Scholar
  12. [PSW] G. Peskir, D. Schneider and M. Weber,Randomly weighted series of contractions in Hilbert spaces, Aarhus Preprint Series no 5 (Danemark), 1994; to appear in Mathematica Scandinavica (1996).Google Scholar
  13. [RW] J. Rosenblatt and M. Wierdl,Fourier analysis and almost everywhere convergence, Cours donnés lors de la conférence “Ergodic Theory and its Connections with Harmonic Analysis” London Mathematical Society Lecture Note Series 205, Cambridge University Press, 1993, pp. 3–152.Google Scholar
  14. [S1] D. Schneider,Convergence presque sûre de moyennes ergodiques perturbées, Comptes Rendus de l’Academie des Sciences, Paris, Série I319 (1994), 1201–1206.MATHGoogle Scholar
  15. [S2] D. Schneider,Convergence presque sûr de moyennes ergodiques perturbées par des variables aléatoires, Thèse prépublication I.R.M.A. Strasbourg, 1994.Google Scholar
  16. [SW1] D. Schneider and M. Weber,Weighted averages of contractions along subsequences, à paraître dans les actes de la “Conference on almost everywhere convergence in ergodic and probability theory”, Ohio State University, Columbus, Juin, 1993, de Gruyter, 1996.Google Scholar
  17. [SW2] D. Schneider et M. Weber,Une remarque sur un théorème de Bourgain, Séminaire de probabilité XXVII, Lecture Notes in Mathematics1557, Springer-Verlag, Berlin, 1993, pp. 202–206.CrossRefGoogle Scholar
  18. [SZ] R. Salem and A. Zygmund,Some properties of trigonometric series whose terms have random signs, Acta Mathematica91 (1954), 245–301.MATHCrossRefMathSciNetGoogle Scholar
  19. [Sz.-NF] B. Sz.-Nagy and C. Foias,Harmonic Analysis of Operators on Hilbert Spaces, North-Holland, Amsterdam, 1970.MATHGoogle Scholar
  20. [Th] J.-P. Thouvenot,La convergence presque sûre des moyennes ergodiques suivant certaines sous-suites d’entiers, Séminaire Bourbaki, 42ème année, Vol. 719, 1989–1990.Google Scholar
  21. [Wi] M. Wierdl,Pointwise ergodic theorem along the primes numbers, Israel Journal of Mathematics64 (1988), 315–336.MathSciNetCrossRefGoogle Scholar

Copyright information

© Hebrew University 1997

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique AvancéeUniversité Louis Pasteur et C.N.R.S.Strasbourg CedexFrance

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