Limit theorems for numbers of multiple returns in non-conventional arrays

YURI KIFER*, B. Fayad, B. Hasselblatt, R. Spatzier

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Pages (from-to)1098-1121
Number of pages24
JournalErgodic Theory and Dynamical Systems
Volume42
Issue number3
DOIs
StatePublished - 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

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