On first passage time structure of random walks |
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Authors: | Ushio Sumita Yasushi Masuda |
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Institution: | Graduate School of Management, University of Rochester, Rochester, NY 14627, USA |
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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. |
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Keywords: | birth-death processes discrete time birth-death chains first passage times conditional first passage time complete monotonicity strong unimodality |
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