Identifying words with unexpected frequencies is an important problem in the analysis of long DNA sequences. To solve it, we need an approximation of the distribution of the number of occurrences $N(W)$ of a word $W$. Modeling DNA sequences with $m$-order Markov chains, we use the Chen-Stein method to obtain Poisson approximations for two different counts. We approximate the ``declumped'' count of $W$ by a Poisson variable and the number of occurrences $N(W)$ by a compound Poisson variable. Combinatorial results are used to solve the general case of overlapping words and to calculate the parameters of these distributions.Key words and phrases Compound Poisson approximation, word counts, word periods, Chen-Stein method.