Abstract
We describe the statistics of repetition times of a string of symbols in a stochastic process.
Denote by τ A the time elapsed until the process spells a finite string A and by S A the number of consecutive repetitions of A. We prove that, if the length of the string grows unboundedly, (1) the distribution of τ A , when the process starts with A, is well approximated by a certain mixture of the point measure at the origin and an exponential law, and (2) S A is approximately geometrically distributed. We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and also allow us to get approximations for all the moments of τ A and S A . To obtain (1) we assume that the process is φ-mixing, while to obtain (2) we assume the convergence of certain conditional probabilities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abadi, M.: Exponential approximation for hitting times in mixing processes. Math. Phys. Electron. J. 7, 2 (2001)
Abadi, M.: Sharp error terms and necessary conditions for exponential hitting times in mixing processes. Ann. Probab. 32(1A), 243–264 (2004)
Abadi, M., Galves, A.: Inequalities for the occurrence times of rare events in mixing processes. The state of the art. Markov Processes Relat. Fields 7(1), 97–112 (2001)
Abadi, M., Vaienti, S.: Statistics properties of repetition times. Preprint (2006)
Blom, G., Thorburn, D.: How many random digits are required until given sequences are obtained? J. Appl. Probab. 19, 518–531 (1982)
Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math, vol. 470. Springer, Berlin (1975)
Chazottes, J.-R.: Hitting and returning to non-rare events in mixing dynamical systems. Nonlinearity 16, 1017–1034 (2003)
Doukhan, P.: Mixing. Properties and Examples. Lecture Notes in Statistics, vol. 85. Springer, Berlin (1995)
Galves, A., Schmitt, B.: Inequalities for hitting times in mixing dynamical systems. Random Comput. Dyn. 5, 337–348 (1997)
Hirata, M.: Poisson law for Axiom A diffeomorphism. Ergod. Theory Dyn. Syst. 13, 533–556 (1993)
Hirata, M., Saussol, B., Vaienti, S.: Statistics of return times: a general framework and new applications. Commun. Math. Phys. 206, 33–55 (1999)
Kac, M.: On the notion of recurrence in discrete stochastic processes. Bull. Am. Math. Soc. 53, 1002–1010 (1947)
Liverani, C., Saussol, B., Vaienti, S.: Conformal measures and decay of correlations for covering weighted systems. Ergod. Theory Dyn. Syst. 18, 1399–420 (1998)
Ornstein, D., Weiss, B.: Entropy and data compression schemes. IEEE Trans. Inf. Theory 39(1), 78–83 (1993)
Stefanov, V.: The intersite distances between pattern occurrences in strings generated by general discrete- and continuous-time models. An algorithmic approach. J. Appl. Probab. 40, 881–892 (2003)
Walters, P.: Ruelle’s operator theorem and g-measures. Trans. Am. Math. Soc. 214, 375–387 (1975)
Wyner, A., Ziv, J.: Some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression. IEEE Trans. Inf. Theory 35(6), 1250–1258 (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abadi, M., Vergne, N. Sharp Error Terms for Return Time Statistics under Mixing Conditions. J Theor Probab 22, 18–37 (2009). https://doi.org/10.1007/s10959-008-0199-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-008-0199-x