## The Annals of Applied Probability

### Saddlepoint approximations and nonlinear boundary crossing probabilities of Markov random walks

#### 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

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

Digital Object Identifier
doi:10.1214/aoap/1050689586

Mathematical Reviews number (MathSciNet)
MR1970269

Zentralblatt MATH identifier
1029.60058

#### 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

#### References

• ARNDT, K. (1980). Asy mptotic properties of the distribution of the supremum of a random walk on a Markov chain. Theory Probab. Appl. 25 309-324.
• BHATTACHARy A, R. N. and RANGA RAO, R.(1976). Normal Approximations and Asy mptotic Expansions. Wiley, New York.
• BOROVKOV, A. A. and ROGOZIN, B. A. (1965). On the multidimensional central limit theorem. Theory Probab. Appl. 10 55-62.
• CHAN, H. P. and LAI, T. L. (2000). Asy mptotic approximations for error probabilities of sequential or fixed sample size tests in exponential families. Ann. Statist. 28 1638-1669.
• CHAN, H. P. and LAI, T. L. (2002). Boundary crossing probabilities for scan statistics and their applications to change-point detection. Methodol. Comput. Appl. Probab. To appear.
• DANIELS, H. E. (1954). Saddlepoint approximation in statistics. Ann. Math. Statist. 25 631-650.
• DE ACOSTA, A. and NEY, P. (1998). Large deviation lower bounds for arbitrary additive functionals of a Markov chain. Ann. Probab. 26 1660-1682.
• DINWOODIE, I. H. (1993). Identifying a large deviation rate function. Ann. Probab. 21 216-231.
• FELLER, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
• GRAY, A. (1990). Tubes. Wesley, New York.
• HIRSCH, M. (1976). Differential Topology. Springer, New York.
• HÖGLUND, T. (1974). Central limit theorems and statistical inference for finite Markov chains. Z. Wahrsch. Verw. Gebiete 29 123-151.
• HÖGLUND, T. (1991). The ruin problem for finite Markov chains. Ann. Probab. 19 1298-1310.
• IGLEHART, D. L. (1972). Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43 627-635.
• ILTIS, M. (1995). Sharp asy mptotics of large deviations in Rd. J. Theoretical Probab. 8 501-524.
• ISCOE, I., NEY, P. and NUMMELIN, E. (1985). Large deviations of uniformly recurrent Markov additive processes. Adv. in Appl. Math. 6 373-412.
• JENSEN, J. L. (1991). Saddlepoint expansions for sums of Markov dependent variables on a continuous state space. Probab. Theory Related Fields 89 181-199.
• JENSEN, J. L. (1995). Saddlepoint Approximations. Oxford Univ. Press.
• KARLIN, S., DEMBO, A. and KAWABATA, T. (1990). Statistical composition of high scoring segments from molecular sequences. Ann. Statist. 18 571-581.
• MEy N, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.
• NEY, P. and NUMMELIN, E. (1987). Markov additive processes. I. Eigenvalues properties and limit theorems, and II. Large deviations. Ann. Probab. 15 561-609.
• PITMAN, J. W. (1975). An identity for stopping times of a Markov process. In Studies in Probability and Statistics (E. J. Williams, ed.) 41-57. North-Holland, Amsterdam.
• SIEGMUND, D. (1988). Approximate tail probabilities for the maxima of some random fields. Ann. Probab. 16 487-501.
• SIEGMUND, D. and VENTRAKAMAN, E. (1995). Using the generalized likelihood ratio statistics for sequential detection of a change-point. Ann. Statist. 23 255-271.
• STONE, C. (1965). Local limit theorem for nonlattice multidimensional distribution functions. Ann. Math. Statist. 36 546-551.
• STANFORD, CALIFORNIA 94305 E-MAIL: lait@stat.stanford.edu