# On the longest gap between power-rate arrivals

## TL;DR

This paper analyzes the asymptotic distribution of the longest gap in an inhomogeneous Poisson process with a power-law rate, revealing a Gumbel limit and a specific growth rate of the gap.

## Contribution

It establishes the limiting distribution of the longest gap in an inhomogeneous Poisson process with power-law rate functions, extending results to include slowly varying terms.

## Key findings

- Longest gap's distribution converges to Gumbel distribution.
- The longest gap's length is asymptotically proportional to (t / log t) times an exponential variable.
- Results are extended to include slowly varying rate functions.

## 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 $\lambda_t$.