Frequency of Pattern Occurences in a (DNA) Sequence

Consider a given pattern H and a random text T oflength n. We assltme that consecutive symbols in the texl are generated either independently or with a Markovian dependency, i.e., we stItely both the so called Bernoulli model and the Markovian model. OUf goal is to assess the limiting distribution of the frequency of the pattern occurrences ln a random sequence. Overlapping copies of a pattern are counted separately! We prove that the number of pattern occurrences tends to a normal distribution, and we derive explicit and asymptotic formulas for the mean and the variance of the pattern occurrence. During the course of the derivation we compute the probab·llity of exactly r occurrences of H in the text T. We derive the generating function of tIllS probability, and using an analytical technique we derive in a uniform manner all results announced above. Applications of these results range from wireless communications to approximate pattern matching, molecular biology, games, codes, and stock market analysis. These findings are of particular interest to molecular biology problems such as finding patterns with unexpected (high or low) frequencies (the so called contrast words) and gene recognition.