On Pattern Occurrences in a Random Text

Consider a given pattern H and a random text T of length n. We assume that symbols in the text occur independently, and various symbols have different probabilities of occurrence (l.e., the so called asymmetric Bernoulli modeQ. We are concerned with the probability of exactly T occurrences of H in the text T. We derive the generating function of this probability, and show that asymptotically it behaves as anrpfi-r-l, where a is an explicitly computed constant, and PH < 1 is the root of an equation depending on the structure of the pattern. We then extend these findings to random patterns.