### 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.

Original language | English |
---|---|

Pages (from-to) | 133-147 |

Number of pages | 15 |

Journal | Stochastic Processes and their Applications |

Volume | 20 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1985 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Mathematics(all)
- Modelling and Simulation
- Statistics and Probability

### Cite this

*Stochastic Processes and their Applications*,

*20*(1), 133-147. https://doi.org/10.1016/0304-4149(85)90021-3

**On first passage time structure of random walks.** / Sumita, Ushio; Masuda, Yasushi.

Research output: Contribution to journal › Article

*Stochastic Processes and their Applications*, vol. 20, no. 1, pp. 133-147. https://doi.org/10.1016/0304-4149(85)90021-3

}

TY - JOUR

T1 - On first passage time structure of random walks

AU - Sumita, Ushio

AU - Masuda, Yasushi

PY - 1985

Y1 - 1985

N2 - 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.

AB - 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.

KW - birth-death processes

KW - complete monotonicity

KW - conditional first passage time

KW - discrete time birth-death chains

KW - first passage times

KW - PF

KW - strong unimodality

UR - http://www.scopus.com/inward/record.url?scp=0039369748&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039369748&partnerID=8YFLogxK

U2 - 10.1016/0304-4149(85)90021-3

DO - 10.1016/0304-4149(85)90021-3

M3 - Article

AN - SCOPUS:0039369748

VL - 20

SP - 133

EP - 147

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 1

ER -