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September 2005 Hitting and return times in ergodic dynamical systems
N. Haydn, Y. Lacroix, S. Vaienti
Ann. Probab. 33(5): 2043-2050 (September 2005). DOI: 10.1214/009117905000000242

Abstract

Given an ergodic dynamical system (X,T,μ), and UX measurable with μ(U)>0, let μ(UU(x) denote the normalized hitting time of xX to U. We prove that given a sequence (Un) with μ(Un)→0, the distribution function of the normalized hitting times to Un 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 , and that in the converging case, $$(⋄)\qquad F(t)=\int_{0}^{t}\bigl(1-\tilde{F}(s)\bigr)\,ds,\qquad t\ge0.$$ 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.

Citation

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N. Haydn. Y. Lacroix. S. Vaienti. "Hitting and return times in ergodic dynamical systems." Ann. Probab. 33 (5) 2043 - 2050, September 2005. https://doi.org/10.1214/009117905000000242

Information

Published: September 2005
First available in Project Euclid: 22 September 2005

zbMATH: 1130.37305
MathSciNet: MR2165587
Digital Object Identifier: 10.1214/009117905000000242

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

Keywords: asymptotic distribution , hitting , Kac , return times

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 5 • September 2005
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