Limiting behaviours of the longest gaps between occurrence epochs in poisson processes

Abstract

This study concerns the limiting behaviours of the longest gap, between epochs in Poisson processes up to time t using large deviation theory. This theory is an area of probability theory which provides tools for studying the probabilistic behaviours of rare events which plays a fundamental role in risk analysis. This gap was considered to be a continuous analog of the longest run in the discrete scenario of the independent and identically distributed (i.i.d.) Bernoulli sequence up to n: The purpose was to study its large deviation probabilities and large deviation principles (LDPs). To achieve this purpose, the following were done. Firstly, the global estimation for the distribution function of was derived with the help of Slivnyak’s formula of Palm Theorem. The derived global estimation provided a very important tool in the subsequent studies such as the derivation of deviation probabilities and hence the large deviation principles of Secondly, since the law of large numbers for in a homogenous Poisson process with unit intensity had been established in the forms of and the large deviation probabilities for the specific convergence speeds of and were derived with the help of the derived global estimation, Fenchel-Legendre transform for computing the rate function and Gärtner-Ellis Theorem for establishing the existence of the large deviation principles. The results showed that L(t) satisfied two large deviation principles with an exponential and power rate function for the specific speeds and respectively. Finally, the study considered a general convergence speed such that for The general large deviation principle was derived with the help of Bryc’s inverse Varadhan Lemma. The result showed that obeys LDP with a general speed and a general rate function. In the derivation of the global estimation and the subsequent application, constant intensity assumption was made. This makes the study only applicable to homogeneous Poisson processes. Cases of varying intensity functions in inhomogeneous Poisson processes therefore present future area of research opportunity.