Laplace-transform asymptotics of longest gaps in Poisson processes

Abstract

In a Poisson process with constant rate (also known as homogeneous Poisson process) with exponential inter-arrival (waiting time to the next event) time, the longest/largest gap, L(t) which is the maximal inter-arrival time is considered in this paper. The aim was to establish the Laplace transform asymptotic and the large deviation principles related to the longest gap L(t) between epochs of arrival in a homogeneous Poisson process. The result of the investigation suggest two natural and different large deviation principles for the longest gap with two distinct rate functions and speeds.