Maxima of Moving Sums in a Poisson Random Field

TL;DR

This paper investigates the tail probabilities of moving sums in a marked Poisson random field, introducing methods to compute and bound these probabilities with improved expressions and potential applications to continuous processes.

Contribution

It develops new techniques for evaluating extremal tail probabilities of moving sums in Poisson fields, including more tractable expressions and bounds for asymptotic constants.

Findings

Derived asymptotic tail probability expressions

Provided computable bounds for these probabilities

Extended methods to continuous processes

Abstract

The extremal tail probabilities of moving sums in a marked Poisson random field is examined here. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. Change of measure and analysis of local random fields are used to provide tail probabilities. The asymptotic constants are initially expressed in a form that seems hard to evaluate and do not seem to provide any additional information on the properties of the constants. A more sophisticated approach is then undertaken giving rise to an expression that is not only neater but also able to provide computable bounds. The technique used to obtain this constant can also be modified to work on continuous processes.