Poisson approximation for the non-overlapping appearances of several words in Markov chains (Q2757073)

Let \(X_1,\dots,X_n\), be a sequence of letters from a stationary Markov chain on a finite alphabet, \(A\) a finite set of words, and \(E_m\) the event that a word from the set \(A\) appears in the \(X\)-sequence with first letter at position \(m\). Let \(N\) be the number of non-overlapping (competing renewal) \(E\)-events occurring in the \(X\)-sequence.NEWLINENEWLINENEWLINEA bound is given for the total variation distance between the distribution of \(N\) and the Poisson distribution with the same mean. The arguments are based on the Stein-Chen method and the combinatorics of words.