On approximation of functions satisfying defective renewal equations

TL;DR

This paper develops a gamma-type operator approximation method for functions satisfying defective renewal equations, common in applied probability, analyzing conditions for optimal convergence and applying results to ruin probabilities with gamma mixtures.

Contribution

Introduces a gamma-type operator approximation for defective renewal functions using Laplace transforms, with convergence analysis and applications to risk models.

Findings

Established conditions for optimal uniform convergence

Derived approximation formulas using Laplace transforms

Applied methods to ruin probability with gamma claim mixtures

Abstract

Functions satisfying a defective renewal equation arise commonly in applied probability models. Usually these functions don't admit a explicit expression. In this work we consider to approximate them by means of a gamma-type operator given in terms of the Laplace transform of the initial function. We investigate which conditions on the initial parameters of the renewal equation give optimal order of uniform convergence in the approximation. We apply our results to ruin probability in the classical risk model, paying special attention to mixtures of gamma claim amounts.