Abstract
We study the asymptotic behavior of ergodic averages of expectation of some randomly selected group of unitary operators, showing the mean convergence when the sequence of selectors is a ℤd-valued random walk. We make use of the spectral decomposition of the unitary group to investigate the more difficult problem of almost sure convergence, and provide sufficient spectral conditions which carry out the almost everywhere convergence of these means when the sequence of selectors is a ℤd-valued random walk satisfying some integrability conditions. We also show that this condition is optimal for d = 1, and deduce a speed of convergence for these averages using a Rademacher-Menchoff theorem on orthogonal series.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Alexits, Problems in the Convergence of the Orthogonal Series (Pergamon Press, New York-London, 1961).
W. Ambrose, Duke. Math. Journal. 11, 589 (1944).
F. Boukhari and M. Weber, Almost sure convergence of sequences of randomly selected ℂd-actions, unpublished work (2000).
F. Boukhari and M. Weber, Illinois J. Math. 46, 1 (2002).
D. Burkholder, Trans. Am. Math. Soc. 104(1), 123 (1962).
S. Durand and D. Schneider, Ergodic. Theory Dynam. Systems. 23, 1059 (2003).
C. Gamet, Théorèmes de convergence en moyennes et entropie métrique en théorie ergodique, Thèse de l’Université de Strasbourg. prépub. I. R.M. A (1996).
C. Gamet and D. Schneider, Ann. Inst. H. Poincaré Probab. Statist. 33(2), 269 (1997).
V. F. Gaposhkin, Theor. Probab. Appl. 22, 286 (1977).
V. F. Gaposhkin, Functional. Anal. Appl. 15, 18 (1981).
R. L. Jones, R. Kaufman, J. M. Rosenblatt, and M. Wierdl, Ergodic TheoryDynam. Systems. 18, 889 (1998).
U. Krengel, Ergodic Theorems (Walters de Gruyter & Co, Berlin, 1985).
M. Lacey, K. Peterson, D. Rudolph, and M. Wierdl, Ann. Inst. H. Poincaré Probab. Statist. 30(3), 353 (1994).
L. Lifshits and M. Weber, Math. Scand. 1 86, 75 (2000).
J. M. Rosenblatt and M. Wierdl, in Proc. Conf. on Ergodic Theory Alexandria, Egypt, 1993 (Cambridge University Press, 1994).
W. Rudin, Fourier analysis on groups (John Wiley & Sons, 1990).
W. Rudin, Functional analysis (McGraw-Hill, Inc, 1991).
D. Schneider, Israël J. Math. 101, 157 (1997).
D. Schneider, Ann. Inst. H. Poincaré. Probab. Statist. 36(5), 617 (2000).
F. Spitzer, Principles of random walk (Springer, 2001).
M. Weber, Dynamical Systems and Processes (EuropeanMathematical Society, 2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Submitted by A. M. Elizarov
Research partially supported by PNR project Y/U13/107, Algeria and the Laboratoire de Statistique et Modélisations Aléatoires.
Rights and permissions
About this article
Cite this article
Boukhari, F. On the almost sure convergence of some ergodic means. Lobachevskii J Math 35, 185–197 (2014). https://doi.org/10.1134/S1995080214030068
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080214030068