On Pattern Frequency Occurrences In A Markovian Sequence?

Consider a given pattern H and a random text T generated by a Markovian source. We study the frequency of pattern occurrences in a random text when overlapping copies of the pattern are counted separately. We present exact and asymptotic formulae for all moments (including the variance), and probability of r pattern occurrences for three different regions of r, namely: (i) r = O(1), (ii) central limit regime, and (iii) large deviations regime. In order to derive these results, we first construct some language expressions that characterize pattern occurrences which are later translated into generating functions. Finally, we use analytical methods to extract asymptotic behaviors of the pattern frequency. Applications of these results include molecular biology, source coding, synchronization, wireless communications, approximate pattern matching, game theory, and stock market analysis. These findings are of particular interest to information theory (e.g., second-order properties of the re..