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arXiv · 2501.18329

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

El'mira Yu. Kalimulina·Galina A. Zverkina
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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.

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El'mira Yu. Kalimulina, Galina A. Zverkina. 2026-03-20. On Lorden's Inequality and Renewal-Type Processes with Dependent Inter-renewal Times. https://arxiv.org/abs/2501.18329

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