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January 2004 Sharp error terms and neccessary conditions for exponential hitting times in mixing processes
Miguel Abadi
Ann. Probab. 32(1A): 243-264 (January 2004). DOI: 10.1214/aop/1078415835


We prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in $\alpha$-mixing processes with exponential decay, $\phi$-mixing processes with a summable function $\phi$ and for general $\psi$-mixing processes with a finite alphabet. In the first case the bound is uniform as a function of the measure of the event. In the last two cases the bound depends also on the time scale t. This allows us to get further statistical properties as the ratio convergence of the expected hitting time and the expected return time. A uniform bound is a consequence. We present an example that shows that this bound is sharp. We also prove that second moments are not necessary for having the exponential law. Moreover, we prove a necessary condition for having the exponential limit law.


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Miguel Abadi. "Sharp error terms and neccessary conditions for exponential hitting times in mixing processes." Ann. Probab. 32 (1A) 243 - 264, January 2004.


Published: January 2004
First available in Project Euclid: 4 March 2004

zbMATH: 1045.60018
MathSciNet: MR2040782
Digital Object Identifier: 10.1214/aop/1078415835

Primary: 37A50 , 60F05 , 60G10

Keywords: exponential approximation , hitting times , mixing processes , Rare event , repetition times

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 1A • January 2004
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