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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Abstract

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

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

Dates
First available in Project Euclid: 18 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1050689586

Digital Object Identifier
doi:10.1214/aoap/1050689586

Mathematical Reviews number (MathSciNet)
MR1970269

Zentralblatt MATH identifier
1029.60058

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aoap/1050689586


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