COMPOUND POISSON AND POISSON-PROCESS APPROXIMATIONS FOR OCCURRENCES OF MULTIPLE WORDS IN MARKOV-CHAINS

We derive a Poisson process approximation for the occurrences of clump s of multiple words and a compound Poisson process approximation for t he number of occurrences of multiple words in a sequence of letters ge nerated by a stationary Markov chain. Using the Chen-Stein method, we provide a bound on the error in the approximations. For rare words, th ese errors tend to zero as the length of the sequence increases to inf inity. Modeling a DNA sequence as a stationary Markov chain, we show a s an application that the compound Poisson approximation is efficient for the number of occurrences of rare stem-loop motifs.