On first passage time structure of random walks

Abstract

For continuous time birth-death processes on {0,1,2,…}, the first passage time T + n from n to n + 1 is always a mixture of ( n + 1) independent exponential random variables. Furthermore, the first passage time T 0, n+1 from 0 to ( n + 1) is always a sum of ( n + 1) independent exponential random variables. The discrete time analogue, however, does not necessarily hold in spite of structural similarities. In this paper, some necessary and sufficient conditions are established under which T + n and T 0, n+1 for discrete time birth-death chains become a mixture and a sum, respectively, of ( n + 1) independent geometric random variables on {1,2,…};. The results are further extended to conditional first passage times.