On probabilities associated with the minimum distance between events of a Poisson process in a finite interval, and Erratum/Addendum to it

Abstract

Original paper: We revisit the probability that any two consecutive events in a Poisson process N on [0,t] are separated by a time interval which is greater than s(<t) (a particular scan statistic probability), and the closely related probability (recently introduced by Todinov [8], who denotes it as p_MFFOP) that before any event of N in [0,t] there exists an event-free interval greater than s. Both probabilities admit simple explicit expressions, which, however, become intractable for very large values of t/s. Our main objective is to demonstrate that these probabilities can be approximated extremely well for large values of t/s by some very tractable and attractive expressions (actually, already for t larger than a few multiples of s). Erratum/Addendum: In this addendum, we further discuss numerical issues concerning the computation of the probability denoted ϕ(s,t;λ) in the original paper. In particular, and most importantly, we correct the naive claim made in the abstract of the original paper that the explicit expression of ϕ(s,t;λ) becomes intractable for very large values of t/s; rather, we show that it may not be applicable for large values of λ*t.