# Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences

ESAIM: Probability and Statistics (2010)

- Volume: 14, page 435-455
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topPudlo, Pierre. "Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences." ESAIM: Probability and Statistics 14 (2010): 435-455. <http://eudml.org/doc/250827>.

@article{Pudlo2010,

abstract = {
To establish lists of words with unexpected frequencies in long sequences,
for instance in a molecular biology context, one needs to
quantify the exceptionality of families of word frequencies in random
sequences. To this aim, we
study large deviation probabilities of multidimensional word counts
for Markov and hidden Markov models.
More specifically, we compute local Edgeworth expansions
of arbitrary degrees for multivariate partial sums of lattice valued
functionals of finite Markov chains. This yields sharp approximations of
the associated large deviation probabilities.
We also provide detailed simulations. These exhibit in particular
previously unreported periodic oscillations, for which we provide
theoretical explanations.
},

author = {Pudlo, Pierre},

journal = {ESAIM: Probability and Statistics},

keywords = {Markov chains; hidden Markov models; large deviations;
edgeworth expansions; protein and DNA sequences; Edgeworth expansions},

language = {eng},

month = {12},

pages = {435-455},

publisher = {EDP Sciences},

title = {Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences},

url = {http://eudml.org/doc/250827},

volume = {14},

year = {2010},

}

TY - JOUR

AU - Pudlo, Pierre

TI - Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences

JO - ESAIM: Probability and Statistics

DA - 2010/12//

PB - EDP Sciences

VL - 14

SP - 435

EP - 455

AB -
To establish lists of words with unexpected frequencies in long sequences,
for instance in a molecular biology context, one needs to
quantify the exceptionality of families of word frequencies in random
sequences. To this aim, we
study large deviation probabilities of multidimensional word counts
for Markov and hidden Markov models.
More specifically, we compute local Edgeworth expansions
of arbitrary degrees for multivariate partial sums of lattice valued
functionals of finite Markov chains. This yields sharp approximations of
the associated large deviation probabilities.
We also provide detailed simulations. These exhibit in particular
previously unreported periodic oscillations, for which we provide
theoretical explanations.

LA - eng

KW - Markov chains; hidden Markov models; large deviations;
edgeworth expansions; protein and DNA sequences; Edgeworth expansions

UR - http://eudml.org/doc/250827

ER -

## References

top- C. Andriani and P. Baldi, Sharp estimates of deviations of the sample mean in many dimensions. Ann. Inst. H. Poincaré Probab. Statist.33 (1997) 371–385. Zbl0882.60022
- R.R. Bahadur and R.R. Rao, On deviations of the sample mean. Ann. Math. Statist.31 (1960) 1015–1027. Zbl0101.12603
- P. Barbe and M. Broniatowski, Large-deviation probability and the local dimension of sets, in Proceedings of the 19th Seminar on Stability Problems for Stochastic Models, Vologda, 1998, Part I. (2000), Vol. 99, pp. 1225–1233. Zbl0962.60011
- N.R. Chaganty and J. Sethuraman, Strong large deviation and local limit theorems. Ann. Probab.21 (1993) 1671–1690. Zbl0786.60026
- S. Datta and W.P. McCormick, On the first-order Edgeworth expansion for a Markov chain. J. Multivariate Anal.44 (1993) 345–359. Zbl0770.60023
- A. Dembo and O. Zeitouni, Large deviations techniques and applications. Volume 38 of Appl. Math. (New York). Second edition. Springer-Verlag, New York (1998). Zbl0896.60013
- P. Flajolet, W. Szpankowski and B. Vallée, Hidden word statistics. J. ACM53 (2006) 147–183 (electronic). Zbl1316.68111
- M. Iltis, Sharp asymptotics of large deviations in Rd. J. Theoret. Probab.8 (1995) 501–522. Zbl0831.60042
- M. Iltis, Sharp asymptotics of large deviations for general state-space Markov-additive chains in Rd. Statist. Probab. Lett.47 (2000) 365–380. Zbl0988.60012
- I. Iscoe, P. Ney and E. Nummelin, Large deviations of uniformly recurrent Markov additive processes. Adv. Appl. Math.6 (1985) 373–412. Zbl0602.60034
- J.L. Jensen, Saddlepoint approximations. The Clarendon Press Oxford University Press, New York (1995). Zbl1274.62008
- V. Kargin, A large deviation inequality for vector functions on finite reversible Markov chains. Ann. Appl. Probab.17 (2007) 1202–1221. Zbl1131.60067
- K. Knopp, Theory of Functions, Part I. Elements of the General Theory of Analytic Functions. Dover Publications, New York (1945).
- I. Kontoyiannis and S.P. Meyn, Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab.13 (2003) 304–362. Zbl1016.60066
- C.A. León and F. Perron, Optimal Hoeffding bounds for discrete reversible Markov chains. Ann. Appl. Probab.14 (2004) 958–970. Zbl1056.60070
- M.E. Lladser, M.D. Betterton and R. Knight, Multiple pattern matching: a Markov chain approach. J. Math. Biol.56 (2008) 51–92. Zbl1147.65005
- B. Mann, Berry-Esseen Central Limit Theorems For Markov Chains. Ph.D. thesis, Harvard University, 1996.
- H.D. Miller, A convexivity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist.32 (1961) 1260–1270. Zbl0108.15101
- P. Ney, Dominating points and the asymptotics of large deviations for random walk on Rd. Ann. Probab.11 (1983) 158–167. Zbl0503.60035
- P. Ney and E. Nummelin, Markov additive processes, Part I. Eigenvalue properties and limit theorems. Ann. Probab.15 (1987) 561–592. Zbl0625.60027
- P. Nicodème, B. Salvy and P. Flajolet, Motif statistics. In Algorithms – ESA '99, Prague. Lect. Notes Comput. Sci.1643. Springer, Berlin (1999), pp 194–211. Zbl0944.92013
- G. Nuel, Numerical solutins for Patterns Statistics on Markov chains. Stat. Appl. Genet. Mol. Biol.5 (2006).
- G. Nuel, Pattern Markov chains: optimal Markov chain embedding through deterministic finite automata. J. Appl. Probab.45 (2008) 226–243. Zbl1142.65010
- R Development Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2003). ISBN 3-900051-00-3.
- M. Régnier, A unified approach to word occurrence probabilities. Discrete Appl. Math.104 (2000) 259–280, Combinatorial molecular biology. Zbl0987.92017
- M. Régnier and A. Denise, Rare events and conditional events on random strings. Discrete Math. Theor. Comput. Sci.6 (2004) 191–213 (electronic). Zbl1059.05008
- M. Régnier and W. Szpankowski, On pattern frequency occurrences in a Markovian sequence. Algorithmica22 (1998) 631–649. Zbl0918.68108
- G. Reinert, S. Schbath and M.S. Waterman, Applied Combinatorics on Words. In Encyclopedia of Mathematics and its Applications, Vol. 105, chap. Statistics on Words with Applications to Biological Sequences. Cambridge University Press (2005).
- S. Robin and J.-J. Daudin, Exact distribution of word occurrences in a random sequence of letters. J. Appl. Probab.36 (1999) 179–193. Zbl0945.60008
- E. Roquain and S. Schbath, Improved compound Poisson approximation for the number of occurrences of any rare word family in a stationary Markov chain. Adv. Appl. Probab.39 (2007) 128–140. Zbl1109.62012
- S. Schbath, Compound Poisson approximation of word counts in DNA sequences. ESAIM: PS1 (1997) 1–16. Zbl0869.60067
- D. Serre, Matrices, volume 216 of Graduate Texts Math.. Springer-Verlag, New York (2002). Theory and applications, translated from the 2001 French original.
- V.T. Stefanov, S. Robin and S. Schbath, Waiting times for clumps of patterns and for structured motifs in random sequences. Discrete Appl. Math.155 (2007) 868–880. Zbl1112.60055

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.