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 ${\lambda }_t$.

19 pages