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May 1999 The asymptotics of waiting times between stationary processes, allowing distortion
Amir Dembo, Ioannis Kontoyiannis
Ann. Appl. Probab. 9(2): 413-429 (May 1999). DOI: 10.1214/aoap/1029962749

Abstract

Given two independent realizations of the stationary processes $\mathbf{X} = {X_n;n \geq 1}$ and $\mathbf{Y} = {Y_n;n \geq 1}$, our main quantity of interest is the waiting time $W_n(D)$ until a D-close version of the initial string $(X_1, X_2,\dots, X_n)$ first appears as a contiguous substring in $(Y_1, Y_2, Y_3,\dots)$, where closeness is measured with respect to some "average distortion" criterion.

We study the asymptotics of $W_n(D)$ for large n under various mixing conditions on X and Y. We first prove a strong approximation theorem between $\logW_n(D)$ and the logarithm of the probability of a D-ball around $(X_1, X_2,\dots, X_n)$. Using large deviations techniques, we show that this probability can, in turn, be strongly approximated by an associated random walk, and we conclude that: (i) $n^{-1} \log W_n(D)$ converges almost surely to a constant R determined byan explicit variational problem; (ii) $[\log W_n(D) - R]$, properly normalized, satisfies a central limit theorem, a law of the iterated logarithm and, more generally, an almost sure invariance principle.

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Amir Dembo. Ioannis Kontoyiannis. "The asymptotics of waiting times between stationary processes, allowing distortion." Ann. Appl. Probab. 9 (2) 413 - 429, May 1999. https://doi.org/10.1214/aoap/1029962749

Information

Published: May 1999
First available in Project Euclid: 21 August 2002

zbMATH: 0940.60033
MathSciNet: MR1687410
Digital Object Identifier: 10.1214/aoap/1029962749

Subjects:
Primary: 60F15
Secondary: 60F10 , 94A17

Keywords: almost sure invariance principle , large deviations , Relative entropy , string matching , strong approximation , Waiting times

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.9 • No. 2 • May 1999
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