On the approximate pattern occurrences in a text

Consider a given pattern H and a random text T generated randomly according to the Bernoulli model. We study the frequency of approximate occurrences of the pattern H in a random text when overlapping copies of the approximate pattern are counted separately. We provide exact and asymptotic formulae for mean, variance and probability of occurrence as well as asymptotic results including the central limit theorem and large deviations. Our approach is combinatorial: we first construct some language expressions that characterize pattern occurrences which are translated into generating functions, and finally we use analytical methods to extract asymptotic behaviours of the pattern frequency. Applications of these results include molecular biology, source coding, synchronization, wireless communications, approximate pattern matching, games, and stock market analysis. These findings are of particular interest to information theory (e.g., second-order properties of the relative frequency), and molecular biology problems (e.g., finding patterns with unexpected high or low frequencies, and gene recognition).