Open Access
February 2019 On the longest gap between power-rate arrivals
Søren Asmussen, Jevgenijs Ivanovs, Johan Segers
Bernoulli 25(1): 375-394 (February 2019). DOI: 10.3150/17-BEJ990


Let $L_{t}$ be the longest gap before time $t$ in an inhomogeneous Poisson process with rate function $\lambda_{t}$ proportional to $t^{\alpha-1}$ for some $\alpha\in(0,1)$. It is shown that $\lambda_{t}L_{t}-b_{t}$ has a limiting Gumbel distribution for suitable constants $b_{t}$ and that the distance of this longest gap from $t$ is asymptotically of the form $(t/\log t)E$ for an exponential random variable $E$. The analysis is performed via weak convergence of related point processes. Subject to a weak technical condition, the results are extended to include a slowly varying term in $\lambda_{t}$.


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Søren Asmussen. Jevgenijs Ivanovs. Johan Segers. "On the longest gap between power-rate arrivals." Bernoulli 25 (1) 375 - 394, February 2019.


Received: 1 March 2017; Revised: 1 August 2017; Published: February 2019
First available in Project Euclid: 12 December 2018

zbMATH: 07007211
MathSciNet: MR3892323
Digital Object Identifier: 10.3150/17-BEJ990

Keywords: Gumbel distribution , inhomogeneous Poisson process , Point processes , records , regular variation , weak convergence

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 1 • February 2019
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