Large Deviations and Full Edgeworth Expansions for Finite Markov Chains with Applications to the Analysis of Genomic Sequences

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. Received July 10, 2008. Revised March 3, 2009 and June 15, 2009. Introduction This paper is devoted to the determination of exact asymptotics of the probabilities of large deviations events for multidimensional additive functionals of finite Markov chains. A motivation that arises in molecular biology is the determination of under and over represented words in genomic sequences (DNA, RNA, and proteins), see Reinert et al. [28] for example. Words with unexpected frequencies in genomic sequences are natural candidates to represent biological signals. A well known example is the Chi motif in the sequence of Escherichia coli , namely the word GCTGGTGG, which is massively over represented and which plays a crucial role in the conservation of the genome of this bacterium. Of course, to detect words with unexpected frequencies, one must specify a stochastic model of the sequence. The most usual models are Markovian, that is, either one assumes that the sequence itself is Markov, or one uses hidden Markov models. Since hidden Markov chains can be represented as functionals of Markov chains, and since Markov chains of higher order are projections of simple Markov chains (that is, Markov chains of order 1) defined on product spaces, functionals of Markov chains and of hidden Markov chains of any order can be viewed as functionals of simple Markov chains. The distribution of the number of visits to a given state by a Markov chain is a much studied subject, in particular in a molecular biology context. For instance, Robin and Daudin [29], Régnier [25] and Stefanov et al. [33] provide exact formulas for these distributions. Nuel [23] turns the occurrences of any pattern over a