Hitting and return times in ergodic dynamical systems

Hitting and return times in ergodic dynamical systems

N. Haydn, Y. Lacroix, and S. Vaienti

Source: Ann. Probab. Volume 33, Number 5 (2005), 2043-2050.

Abstract

Given an ergodic dynamical system (X,T,μ), and U⊂X measurable with μ(U)>0, let μ(U)τ_U(x) denote the normalized hitting time of x∈X to U. We prove that given a sequence (U_n) with μ(U_n)→0, the distribution function of the normalized hitting times to U_n converges weakly to some subprobability distribution F if and only if the distribution function of the normalized return time converges weakly to some distribution function F̃, and that in the converging case,

$(\diamondsuit)\hspace*{66pt}F(t)=\int_{0}^{t}\bigl(1-\tilde{F}(s)\bigr)\,ds,\qquad t\ge0.\hspace*{66pt}$

This in particular characterizes asymptotics for hitting times, and shows that the asymptotics for return times is exponential if and only if the one for hitting times is also.

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Primary Subjects: 37A05, 37A50, 60F05, 28D05

Keywords: Asymptotic distribution; hitting; return times; Kac

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1127395881
Digital Object Identifier: doi:10.1214/009117905000000242
Mathematical Reviews number (MathSciNet): MR2165587
Zentralblatt MATH identifier: 02228512

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