An extension of Hewitt's inversion formula and its application to fluctuation theory

TL;DR

This paper extends Hewitt's inversion formula to analyze fluctuations of random walks with general increments, providing integral representations for key measures useful in queueing and risk management.

Contribution

It introduces an extended inversion theorem for Laplace-Stieltjes transforms and applies it to fluctuation theory, including explicit calculations under rational increment transforms.

Findings

Derived integral representations for fluctuation measures

Extended Hewitt's inversion formula for broader applications

Applicable to queueing and insurance risk models

Abstract

We analyze fluctuations of random walks with generally distributed increments. Integral representations for key performance measures are obtained by extending an inversion theorem of Hewitt [11] for Laplace-Stieltjes transforms. Another important part of the anal- ysis involves the so-called harmonic measures associated to the distribution of the increment of the walk. It is also pointed out that such representations can be explicitly calculated, if one assumes a form of rational structure for the increment transform. Applications include, but are not restricted to, queueing and insurance risk problems.