The Annals of Applied Probability

Local alignment of Markov chains

Niels Richard Hansen

Full-text: Open access

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

Permanent link to this document
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

Keywords
Chen–Stein method extreme value theory large deviations local alignment Markov additive processes Markov chains Poisson approximation

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


Export citation

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.