General large deviations of longest gaps in homogeneous Poisson processes

Abstract

Let (N(t))t≥0 be a homogeneous Poisson process with unit intensity. The longest gap L(t)of this process is the maximal time subinterval between epochs of arrival times up to time t, and it finds applications in the theory of reliability. It has been known since the 1980s that the longest gap obeys the laws of large numbers L(t)/ lnt →1as t →∞almost surely. However large deviation probabilities of the longest gap have not been well developed yet, and in this paper we derive such large deviations with a very general speed. The main tool is to establish a global estimation for the distribution function of the longest gap which is of independent interest.