Abstract
For a Formula Presented-mixing process Formula Presented we consider the number Formula Presented of multiple returns Formula Presented to a set Formula Presented for n until either a fixed number N or until the moment Formula Presented when another multiple return Formula Presented, takes place for the first time where Formula Presented and Formula Presented, Formula Presented are certain functions of n taking on non-negative integer values when n runs from 0 to N. The dependence of Formula Presented on both n and N is the main novelty of the paper. Under some restrictions on the functions Formula Presented we obtain Poisson distributions limits of Formula Presented when counting is until N as Formula Presented and geometric distributions limits when counting is until Formula Presented as Formula Presented. We obtain also similar results in the dynamical systems setup considering a Formula Presented-mixing shift T on a sequence space Formula Presented and studying the number of multiple returns Formula Presented until the first occurrence of another multiple return Formula Presented where Formula Presented are cylinder sets of length n and m constructed by sequences Formula Presented, respectively, and chosen so that their probabilities have the same order.
Original language | English |
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Pages (from-to) | 1098-1121 |
Number of pages | 24 |
Journal | Ergodic Theory and Dynamical Systems |
Volume | 42 |
Issue number | 3 |
DOIs | |
State | Published - 12 Mar 2022 |
Bibliographical note
Publisher Copyright:© The Author(s), 2021. Published by Cambridge University Press
Keywords
- 37D35
- 60F05
- 60J05
- geometric and Poisson distributions
- multiple returns
- non-conventional sums