On first passage time structure of random walks

Ushio Sumita and Yasushi Masuda

Stochastic Processes and their Applications, 1985, vol. 20, issue 1, 133-147

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 T0,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 T0,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.

Keywords: birth-death; processes; discrete; time; birth-death; chains; first; passage; times; conditional; first; passage; time; complete; monotonicity; strong; unimodality; PF[infinity] (search for similar items in EconPapers)
Date: 1985
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