## Annals of Applied Probability

### Local alignment of Markov chains

Niels Richard Hansen

#### Abstract

We consider local alignments without gaps of two independent Markov chains from a finite alphabet, and we derive sufficient conditions for the number of essentially different local alignments with a score exceeding a high threshold to be asymptotically Poisson distributed. From the Poisson approximation a Gumbel approximation of the maximal local alignment score is obtained. The results extend those obtained by Dembo, Karlin and Zeitouni [Ann. Probab. 22 (1994) 2022–2039] for independent sequences of i.i.d. variables.

#### Article information

Source
Ann. Appl. Probab., Volume 16, Number 3 (2006), 1262-1296.

Dates
First available in Project Euclid: 2 October 2006

https://projecteuclid.org/euclid.aoap/1159804981

Digital Object Identifier
doi:10.1214/105051606000000321

Mathematical Reviews number (MathSciNet)
MR2260063

Zentralblatt MATH identifier
1113.60054

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F10: Large deviations

#### Citation

Hansen, Niels Richard. Local alignment of Markov chains. Ann. Appl. Probab. 16 (2006), no. 3, 1262--1296. doi:10.1214/105051606000000321. https://projecteuclid.org/euclid.aoap/1159804981

#### References

• Alsmeyer, G. (2000). The ladder variables of a Markov random walk. Probab. Math. Statist. 20 151–168.
• Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen–Stein method. Ann. Probab. 17 9–25.
• Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Springer, New York.
• Bundschuh, R. (2000). An analytic approach to significance assessment in local sequence alignment with gaps. In RECOMB 00: Proceedings of the Fourth Annual International Conference on Computational Molecular Biology 86–95. ACM Press, New York.
• Çinlar, E. (1975). Introduction to Stochastic Processes. Prentice–Hall Inc., Englewood Cliffs, NJ.
• Dembo, A., Karlin, S. and Zeitouni, O. (1994). Limit distribution of maximal non-aligned two-sequence segmental score. Ann. Probab. 22 2022–2039.
• Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
• den Hollander, F. (2000). Large Deviations. Amer. Math. Soc., Providence, RI.
• Doukhan, P. (1994). Mixing. Springer, New York.
• Grossmann, S. and Yakir, B. (2004). Large deviations for global maxima of independent superadditive processes with negative drift and an application to optimal sequence alignments. Bernoulli 10 829–845.
• Karlin, S. and Dembo, A. (1992). Limit distributions of maximal segmental score among Markov-dependent partial sums. Adv. in Appl. Probab. 24 113–140.
• Kingman, J. F. C. (1961). A convexity property of positive matrices. Quart. J. Math. Oxford Ser. (2) 12 283–284.
• Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, Berlin.
• Mercier, S. (2001). Exact distribution for the local score of one i.i.d. random sequence. J. Comput. Biol. 8 373–380. Available at http://www.liebertonline.com/doi/abs/10.1089/106652701752236197.
• Mercier, S. and Hassenforder, C. (2003). Distribution exacte du score local, cas markovien. C. R. Math. Acad. Sci. Paris 336 863–868.
• O'Cinneide, C. (2000). Markov additive processes and Perron–Frobenius eigenvalue inequalities. Ann. Probab. 28 1230–1258.
• Siegmund, D. and Yakir, B. (2000). Approximate p-values for local sequence alignments. Ann. Statist. 28 657–680.
• Siegmund, D. and Yakir, B. (2003). Correction: “Approximate p-values for local sequence alignments” [Ann. Statist. 28 (2000) 657–680 MR2002a:62140]. Ann. Statist. 31 1027–1031.
• Steele, J. M. (1997). Probability Theory and Combinatorial Optimization. SIAM, Philadelphia.
• Takahata, H. (1981). $L_{\infty}$-bound for asymptotic normality of weakly dependent summands using Stein's result. Ann. Probab. 9 676–683.
• Waterman, M. S., ed. (1995). Introduction to Computational Biology. Chapman and Hall/CRC, Boca Raton, FL.