On Lorden's Inequality and Renewal-Type Processes with Dependent Inter-renewal Times
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Kalimulina, El'mira Yu. Zverkina, Galina A. |
| author_facet | Kalimulina, El'mira Yu. Zverkina, Galina A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18329 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Lorden's Inequality and Renewal-Type Processes with Dependent Inter-renewal Times Kalimulina, El'mira Yu. Zverkina, Galina A. Probability 60K05 60K05, 60K20, 60J05 G.3; F.2.2 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. |
| title | On Lorden's Inequality and Renewal-Type Processes with Dependent Inter-renewal Times |
| topic | Probability 60K05 60K05, 60K20, 60J05 G.3; F.2.2 |
| url | https://arxiv.org/abs/2501.18329 |