On generating functions of waiting time problems for sequence patterns of discrete random variables (Q1293648)

Let \(X_1,X_2,\dots\) be a sequence of independent identically distributed random variables taking values in a countable set \(S= \{0,1,2,\dots\}\). A finite sequence of elements in \(S\) is said to be a pattern. Suppose that a sequence of positive integers \(\{k_i\}^\infty_{i=0}\) is given and set \(P_i= \alpha_{i,1}\alpha_{i,2}\cdots \alpha_{i,k_i}\). Then \(P_i\) is a pattern of length \(k_i\). The author assumes that \(\{P_i\}\cap \{P_j\}= \emptyset\), for \(i\neq j\), and \(\alpha_{i,1}\leq \alpha_{i,2}\leq\cdots\leq \alpha_{i,k}\) for \(i= 0,1,2,\dots\), and denotes by \(E_i\) the event that \(P_i\) occurs. The author employs the method of generalized probability generating functions (gpgf's) of \textit{M. Ebneshahrashoob} and \textit{M. Sobel} [Stat. Probab. Lett. 9, No. 1, 5-11 (1990; Zbl 0695.60016)] to derive the gpgf of the distribution of the waiting time until the \(r\)th occurrence among the events \(\{E_i\}^\infty_{i= 0}\). He also derives the gpf of the distribution of the number of occurrences of the subpattern \(\alpha_1\alpha_2\cdots \alpha_i\) \((m\leq i< k)\) until the first occurrence of the pattern \(\alpha_1\alpha_2\cdots\alpha_k\) in an \(m\)th order Markov chain.