The Annals of Applied Probability

Saddlepoint approximations and nonlinear boundary crossing probabilities of Markov random walks

Hock Peng Chan and Tze Leung Lai

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Saddlepoint approximations are developed for Markov random walks $S_n$ and are used to evaluate the probability that $(j-i) g((S_j - S_i)/(j-i))$ exceeds a threshold value for certain sets of $(i,j)$. The special case $g(x) = x$ reduces to the usual scan statistic in change-point detection problems, and many generalized likelihood ratio detection schemes are also of this form with suitably chosen $g$. We make use of this boundary crossing probability to derive both the asymptotic Gumbel-type distribution of scan statistics and the asymptotic exponential distribution of the waiting time to false alarm in sequential change-point detection. Combining these saddlepoint approximations with truncation arguments and geometric integration theory also yields asymptotic formulas for other nonlinear boundary crossing probabilities of Markov random walks satisfying certain minorization conditions.

Article information

Ann. Appl. Probab., Volume 13, Number 2 (2003), 395-429.

First available in Project Euclid: 18 April 2003

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60F10: Large deviations 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60G60: Random fields 60J05: Discrete-time Markov processes on general state spaces

Markov additive processes large deviation maxima of random fields change-point detection Laplace's method integrals over tubes


Chan, Hock Peng; Lai, Tze Leung. Saddlepoint approximations and nonlinear boundary crossing probabilities of Markov random walks. Ann. Appl. Probab. 13 (2003), no. 2, 395--429. doi:10.1214/aoap/1050689586.

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