On Lorden's Inequality and Renewal-Type Processes with Dependent Inter-renewal Times

TL;DR

This paper extends Lorden's inequality to renewal processes with dependent, non-identically distributed inter-renewal times, providing bounds useful for coupling and convergence analysis in complex stochastic models.

Contribution

It introduces a generalized intensity measure and a Lorden-type bound applicable to dependent and mixed distribution renewal processes, expanding classical renewal theory.

Findings

Provides explicit first-moment bounds for various renewal benchmarks.

Verifies the Lorden constant in Markov-modulated cases.

Identifies second-moment thresholds for Pareto distributions.

Abstract

We consider renewal-type processes whose positive inter-renewal times may be dependent, non-identically distributed, and may have mixed distributions. We introduce a generalised intensity measure extending the classical hazard-rate representation to this setting. Under a two-sided comparison scheme for the inter-renewal laws and an additional renewal-measure domination condition \textnormal{(RD)}, we prove a Lorden-type bound for the forward recurrence time. This bound provides an explicit first-moment input for coupling constructions and, once the remaining coupling parameters are controlled, yields a total-variation estimate. We illustrate the result on exponential, mixed, Markov-modulated, and Pareto benchmarks. In the i.i.d.\ benchmarks, the bound has the correct renewal scale up to a universal factor; in the Markov-modulated benchmark, the explicit Lorden constant is verified while the final convergence consequence remains conditional on \textnormal{(RD)}; and in the Pareto case the construction identifies the natural second-moment threshold for finiteness of the Lorden input.