Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences
Université Montpellier 2, Place E. Bataillon, 34095 Montpellier, Cedex, France; email@example.com
Revised: 3 March 2009
Revised: 15 June 2009
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.
Mathematics Subject Classification: 60J10 / 60F10 / 60J55 / 92D20 / 60F05
Key words: Markov chains / hidden Markov models / large deviations / edgeworth expansions / protein and DNA sequences
© EDP Sciences, SMAI, 2010