On first passage time structure of random walks

Ushio Sumita, Yasushi Masuda

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)133-147
Number of pages15
JournalStochastic Processes and their Applications
Issue number1
Publication statusPublished - 1985 Jul
Externally publishedYes


  • PF
  • birth-death processes
  • complete monotonicity
  • conditional first passage time
  • discrete time birth-death chains
  • first passage times
  • strong unimodality

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics


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