Waiting times for patterns in a sequence of multistate trials (Q2748443)

Let \((X_n,n\geq 1)\) be a sequence of trials taking values in a given finite set \(A\). A simple pattern is defined to be a finite sequence of outcomes from \(A\) and a compound pattern is defined to be a union of distinct simple patterns. Patterns will be denoted by (simple or compound) \({\mathcal E}\) and let \(X_{r,{\mathcal E}}\) be a random variable denoting the waiting time for the \(r\)th occurrence of \({\mathcal E}\). There are a number of papers dealing with the study of \(X_{r,{\mathcal E}}\) and related problems. The present article develops a variation of finite Markov chain embedding method for the derivation of the probability mass and probability generating functions of \(X_{r,{\mathcal E}}\) in the non-overlapping and the overlapping way of counting runs and patterns. Several extensions and generalizations are discussed. NEWLINENEWLINENEWLINEFor related papers see: \textit{D. L. Antzoulakos} [Ann. Inst. Stat. Math. 51, No. 2, 323-330 (1999; Zbl 0949.60085)], \textit{J. C. Fu} [Stat. Sin. 6, No. 4, 957-974 (1996; Zbl 0857.60068)] and \textit{M. V. Koutras} [Ann. Inst. Stat. Math. 49, No. 1, 123-139 (1997; Zbl 0913.60019)].