Moments of Gamma type and the Brownian supremum process area

TL;DR

This paper explores a class of positive random variables with moments linked to Gamma functions, including Brownian supremum process area, providing new insights into their density, asymptotics, and applications.

Contribution

It introduces a framework for analyzing moments of Gamma type, applies it to Brownian supremum process area, and connects it to practical problems like hashing with linear displacement.

Findings

Moments of the Brownian supremum process area are of Gamma type.

Derived series expansions and asymptotic behaviors for related densities.

Established applications to hashing and other stochastic models.

Abstract

We study positive random variables whose moments can be expressed by products and quotients of Gamma functions; this includes many standard distributions. General results are given on existence, series expansion and asymptotics of density functions. It is shown that the integral of the supremum process of Brownian motion has moments of this type, as well as a related random variable occuring in the study of hashing with linear displacement, and the general results are applied to these variables.