Abstract
Given two independent realizations of the stationary processes X=Xn;n≥1 and Y=Yn;n≥1, our main quantity of interest is the waiting time Wn(D) until a D-close version of the initial string (X1,X2,…,Xn) first appears as a contiguous substring in (Y1,Y2,Y3,…), where closeness is measured with respect to some "average distortion" criterion.
We study the asymptotics of Wn(D) for large n under various mixing conditions on X and Y. We first prove a strong approximation theorem between \logWn(D) and the logarithm of the probability of a D-ball around (X1,X2,…,Xn). 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−1logWn(D) converges almost surely to a constant R determined byan explicit variational problem; (ii) [logWn(D)−R], properly normalized, satisfies a central limit theorem, a law of the iterated logarithm and, more generally, an almost sure invariance principle.
Citation
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
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