On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables

Abstract

Let X$_1$, X$_2$, ... be a sequence of independent and identically distributed random variables, which take values in a countable set S = {0, 1, 2, ...}. By a pattern we mean a finite sequence of elements in S. For every i = 0, 1, 2, ..., we denote by P$_i$ = "a$_{i,1}$a$_{i,2}$... a$_{i,k_i }$" the pattern of some length k$_i$, and E$_i$ denotes the event that the pattern P$_i$ occurs in the sequence X$_1$, X$_2$, .... In this paper, we have derived the generalized probability generating functions of the distributions of the waiting times until the r-th occurrence among the events $\{ E_i \} _{i = 0}^\infty$. We also have derived the probability generating functions of the distributions of the number of occurrences of sub-patterns of length l(l < k) until the fiurrence of the pattern of length k in the higher order Markov chain.