Asymptotic results in-over and under-representation of words in DNA

Abstract

Identifying over- and under-represented words is often useful in extracting information of DNA sequences. In this thesis, we shall focus on the words of maximal and minimal occurrences, which will be definitely regarded as over- and under-represented words respectively. We study the tail probabilities of the extrema over a finite set of standard normal random variables by using techniques like Bonferroni's inequalities and Poisson Approximation. We apply similar techniques and the moderate deviations of m-dependent random variables together, and then derive the asymptotic tail probabilities of extrema over a set of word occurrences under M0 model. The statistical distribution of word counts is also studied. We show the asymptotic normality of word counts under both the M0 and M1 models. Finally we use computer simulations to study the tail probabilities of the most frequently and most rarely occurred DNA words under both the M0 and M1 models. The asymptotic results under the M1 model are shown to be similar to those for the M0 model.