The moment generating function of the stopping time for a linearly stopped Poisson process

James DeLucia, H. Vincent Poor

Research output: Contribution to journalArticle

7 Scopus citations

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 \ Xt - ct ∉ (A, B)} are examined, where {Xt; 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 languageEnglish (US)
Pages (from-to)275-292
Number of pages18
JournalCommunications in Statistics. Part C: Stochastic Models
Volume13
Issue number2
DOIs
StatePublished - Jan 1 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

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