Sharp error terms for return time statistics under mixing conditions

TL;DR

This paper analyzes the distribution of return times and repetition counts for strings in a stochastic process under mixing conditions, providing sharp error bounds and moment approximations.

Contribution

It introduces precise error estimates for return time and repetition distributions in phi-mixing processes, extending understanding of their statistical behavior.

Findings

Distribution of T(A) approximated by a mixture of point and exponential laws

S(A) is approximately geometrically distributed

Provided sharp point-wise error bounds for these approximations

Abstract

We describe the statistics of repetition times of a string of symbols in a stochastic process. Denote by T(A) the time elapsed until the process spells the finite string A and by S(A) the number of consecutive repetitions of A. We prove that, if the length of the string grows unbondedly, (1) the distribution of T(A), when the process starts with A, is well aproximated 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 allow to get also approximations for all the moments of T(A) and S(A). To obtain (1) we assume that the process is phi-mixing while to obtain (2) we assume the convergence of certain contidional probabilities.