Higher order return times for $φ$-mixing measures

TL;DR

This work proves that for $\phi$-mixing measures, the limiting distribution of higher-order return times to shrinking target sets is compound Poisson. By developing a convolution-based blocking technique and a generating-function framework, the authors connect multi-step returns to short-block behavior and identify the limiting cluster probabilities $\lambda_k$. The main theorem formalizes the convergence of $Z_{U_n}^{N_n}$ to a compound Poisson law with parameters $t$ and $\lambda_k$ under suitable decay and scaling conditions, extending prior results from first-entry times to higher orders. The results apply across broad dynamical settings, including $\,\alpha$-mixing and Gibbs–Markov systems, and illuminate the role of extremal indices and Kac scaling in return-time statistics. Overall, the paper advances the understanding of Poisson-type limit laws for shrinking targets in ergodic theory and statistical mechanics.

Abstract

In this paper we consider $φ$-mixing measures and show that the limiting return times distribution is compound Poisson distribution as the target sets shrink to a zero measure set. The approach we use generalises a method given by Galves and Schmitt in 1997 for the first entry time to higher orders.

Theorems & Definitions (22)

• Theorem 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• proof : Proof of Theorem \ref{['sequence.theorem']}