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 \(\lambda_tL_t-b_t\) 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 \(\lambda_t\).