## Abstract

First-crossing times of linear boundaries by Poisson processes are considered. In particular, the moment generating functions of stopping times of the form T = inf {t ≥ 0 \ X_{t} - ct ∉ (A, B)} are examined, where {X_{t}; t ≥ 0} is a homogeneous Poisson counting process and -∞ < A < B ≤ ∞. Exact and asymptotic (in B) expressions are obtained by analyzing certain delay-differential equations derived for this moment generating function by Dvoretsky, Kiefer and Wolfowitz. Further results are obtained for the case of a single, lower boundary (i.e., B = ∞) by exploiting results of Zacks who gives a series representation for the stopping-time distribution in this case. These results are used to obtain a closed-form expression for the cumulants of the stopping time in this latter case.

Original language | English (US) |
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Pages (from-to) | 275-292 |

Number of pages | 18 |

Journal | Communications in Statistics. Part C: Stochastic Models |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - 1997 |

## All Science Journal Classification (ASJC) codes

- Modeling and Simulation

## Keywords

- Delay-Differential Equations
- Moment Generating Function
- Poisson Processes
- Sequential Probability Ratio Test
- Stopping Times