The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 13, Number 2 (2003), 395-429.
Saddlepoint approximations and nonlinear boundary crossing probabilities of Markov random walks
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.
Ann. Appl. Probab., Volume 13, Number 2 (2003), 395-429.
First available in Project Euclid: 18 April 2003
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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
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