A Poisson approximation for the number of \(k\)-matches (Q1336933)

Let \(y_ 1,\dots,y_ n\) be independent and identically distributed random variables, with only finitely many possible values. Set \(I_ i = I[y_ i \in \{y_{i-1},\dots,y_{i-k}\}]\) if \(i > k\), \(I_ i = I[y_ i \in \{y_{i - 1}, \dots , y_ 1\}]\) if \(i \leq k\), and let \(X_ n = \sum^ n_{i = 1} I_ i\). The authors use the Stein-Chen method to show how close the distribution of \(X_ n\) is to a Poisson distribution. The results improve those of \textit{B. C. Arnold} [J. Appl. Probab. 9, 841-846 (1972; Zbl 0248.60016)].