On some generalization of Lorden's inequality for renewal processes

TL;DR

This paper extends Lorden's inequality, originally for independent renewal processes, to dependent variables with finite expectations, broadening its applicability in queueing theory and stochastic processes.

Contribution

It generalizes Lorden's inequality to dependent random variables with finite expectations, under certain boundary conditions and integrability constraints.

Findings

Established a version of Lorden's inequality for dependent variables

Proved bounds for expectations of renewal times with dependent variables

Extended the inequality's applicability beyond i.i.d. assumptions

Abstract

In queueing theory, Lorden's inequality can be used for bounds estimation of the moments of backward and forward renewal times. Two random variables called backwards renewal time and forward renewal time for this process are defined. Lorden's inequality it's true for the renewal process, so expectations of backward and forward renewal times are bounded by the relation of expectation of moment of the random variable for any fixed moment of time, where random variables are i.i.d. We generalised and proved a similar result for dependent random variables with finite expectations, some constant C and integrable function Q(s): if X is not independent and have absolutely continuous distribution function which satisfies some boundary conditions, then the analogue of Lorden's inequality for renewal process is true. In August 2021 reviewed version is uploaded.