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

TL;DR

This paper analyzes probabilities related to the minimum distance between events in a Poisson process, providing explicit expressions and approximations for large intervals, and discusses numerical issues in their computation.

Contribution

It introduces accurate approximations for key Poisson process probabilities and clarifies computational limitations for large parameters, enhancing practical applicability.

Findings

Explicit formulas for scan statistic probabilities

Effective approximations for large t/s values

Correction of previous claims on computational intractability

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 \phi(s,t;\lambda) 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 \phi(s,t;\lambda) becomes intractable for very large values of t/s; rather, we show that it may not be applicable for large values of \lambda*t.