## Bernoulli

• Bernoulli
• Volume 25, Number 1 (2019), 375-394.

### On the longest gap between power-rate arrivals

#### Abstract

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 $\lambda_{t}$.

#### Article information

Source
Bernoulli, Volume 25, Number 1 (2019), 375-394.

Dates
Revised: August 2017
First available in Project Euclid: 12 December 2018

https://projecteuclid.org/euclid.bj/1544605250

Digital Object Identifier
doi:10.3150/17-BEJ990

Mathematical Reviews number (MathSciNet)
MR3892323

Zentralblatt MATH identifier
07007211

#### Citation

Asmussen, Søren; Ivanovs, Jevgenijs; Segers, Johan. On the longest gap between power-rate arrivals. Bernoulli 25 (2019), no. 1, 375--394. doi:10.3150/17-BEJ990. https://projecteuclid.org/euclid.bj/1544605250

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