The authors are concerned with \(N\)-letter sequences \(\left( a_{1},\dots ,a_{N}\right) \) where the letters \(a_{i},\;1\leq i\leq N\), are selected from an \(\alpha \)-letter front \(\left\{ 1,2,\dots ,\alpha \right\}\), with the convention that \(a_{j-N}=a_{j}\). First, for any \(r\geq 2\), there is constructed the generating function for the joint distribution of the numbers of times that specified different \(r\)-letter words appear in a sequence of letters. The latter is defined on a ring, and its distribution is chosen as \(\left( r-1\right)\)th-order Markov. Second, the Poisson limit of the resulting distribution of words is obtained explicitly. Then the case of i.i.d. sequences is paid special attention. Correction terms to the Poisson limit are given and, for \(r=2\), the exact distribution is derived. Finally, it it shown how a hidden Markov model fits into the scheme considered.