The Annals of Applied Probability

Local alignment of Markov chains

Niels Richard Hansen

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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


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References

  • Alsmeyer, G. (2000). The ladder variables of a Markov random walk. Probab. Math. Statist. 20 151--168.
    Mathematical Reviews (MathSciNet): MR1785244
  • Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen--Stein method. Ann. Probab. 17 9--25.
    Mathematical Reviews (MathSciNet): MR0972770
    Digital Object Identifier: doi:10.1214/aop/1176991491
    Project Euclid: euclid.aop/1176991491
    Zentralblatt MATH: 0675.60017
  • Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1978607
    Zentralblatt MATH: 1029.60001
  • 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.
    Mathematical Reviews (MathSciNet): MR0380912
  • Dembo, A., Karlin, S. and Zeitouni, O. (1994). Limit distribution of maximal non-aligned two-sequence segmental score. Ann. Probab. 22 2022--2039.
    Mathematical Reviews (MathSciNet): MR1331214
    Digital Object Identifier: doi:10.1214/aop/1176988493
    Project Euclid: euclid.aop/1176988493
    Zentralblatt MATH: 0836.60023
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1619036
    Zentralblatt MATH: 0896.60013
  • den Hollander, F. (2000). Large Deviations. Amer. Math. Soc., Providence, RI.
    Mathematical Reviews (MathSciNet): MR1739680
    Zentralblatt MATH: 0949.60001
  • Doukhan, P. (1994). Mixing. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1312160
  • 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.
    Mathematical Reviews (MathSciNet): MR2093612
    Digital Object Identifier: doi:10.3150/bj/1099579157
    Project Euclid: euclid.bj/1099579157
    Zentralblatt MATH: 1068.60037
  • Karlin, S. and Dembo, A. (1992). Limit distributions of maximal segmental score among Markov-dependent partial sums. Adv. in Appl. Probab. 24 113--140.
    Mathematical Reviews (MathSciNet): MR1146522
    Digital Object Identifier: doi:10.2307/1427732
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0767.60017
  • Kingman, J. F. C. (1961). A convexity property of positive matrices. Quart. J. Math. Oxford Ser. (2) 12 283--284.
    Mathematical Reviews (MathSciNet): MR0138632
    Digital Object Identifier: doi:10.1093/qmath/12.1.283
    Zentralblatt MATH: 0101.25302
  • Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1102015
    Zentralblatt MATH: 0748.60004
  • 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.
    Mathematical Reviews (MathSciNet): MR1990029
    Digital Object Identifier: doi:10.1016/S1631-073X(03)00208-5
    Zentralblatt MATH: 1039.60066
  • O'Cinneide, C. (2000). Markov additive processes and Perron--Frobenius eigenvalue inequalities. Ann. Probab. 28 1230--1258.
    Mathematical Reviews (MathSciNet): MR1797311
    Digital Object Identifier: doi:10.1214/aop/1019160333
    Project Euclid: euclid.aop/1019160333
    Zentralblatt MATH: 1025.15027
  • Siegmund, D. and Yakir, B. (2000). Approximate p-values for local sequence alignments. Ann. Statist. 28 657--680.
    Mathematical Reviews (MathSciNet): MR1792782
    Project Euclid: euclid.aos/1015951993
  • 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.
    Mathematical Reviews (MathSciNet): MR1792782
    Project Euclid: euclid.aos/1015951993
  • Steele, J. M. (1997). Probability Theory and Combinatorial Optimization. SIAM, Philadelphia.
    Mathematical Reviews (MathSciNet): MR1422018
    Zentralblatt MATH: 0916.90233
  • Takahata, H. (1981). $L_\infty$-bound for asymptotic normality of weakly dependent summands using Stein's result. Ann. Probab. 9 676--683.
    Mathematical Reviews (MathSciNet): MR0624695
    Digital Object Identifier: doi:10.1214/aop/1176994375
    Project Euclid: euclid.aop/1176994375
    Zentralblatt MATH: 0465.60033
  • Waterman, M. S., ed. (1995). Introduction to Computational Biology. Chapman and Hall/CRC, Boca Raton, FL.
    Zentralblatt MATH: 0831.92011

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