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Generalised Lorden's Inequality and Convergence Rate Estimates for Non-regenerative Stochastic Models

El'mira Yu. Kalimulina, Galina A. Zverkina

TL;DR

This work addresses the limitation of classical renewal theory for systems with dependent and heterogeneous renewal times by generalising Lorden's inequality through a generalised renewal intensity. It develops a parallel (successful) coupling framework to obtain explicit convergence-rate bounds to a stationary distribution in total variation without requiring explicit generators. The key contribution is a computable bound on the overshoot $B_t$ and an associated convergence-rate estimate for generalised renewal processes, enabling ergodicity results in reliability models and Markov-modulated settings. The approach offers a flexible, efficient alternative for analyzing complex networks and reliability systems where traditional matrix-analytic methods are impractical.

Abstract

We present a generalisation of Lorden's inequality tailored for stochastic systems with dependent and non-identically distributed renewal times modelled as generalised renewal processes. This framework introduces the new concept of a generalised intensity, enabling the analysis of systems where classical renewal theory is inapplicable. The proposed approach is applied to a variety of complex systems, including reliability models, queueing networks and Markov-modulated processes, to establish ergodicity and derive convergence rate estimates. Unlike traditional methods relying on transition matrices or generators, our technique provides an alternative with some rigorous upper bounds, but computationally much more efficient than existent ones. This work offers a unified framework for analysing systems with dependencies and heterogeneities, bridging gaps in classical stochastic modelling.

Generalised Lorden's Inequality and Convergence Rate Estimates for Non-regenerative Stochastic Models

TL;DR

This work addresses the limitation of classical renewal theory for systems with dependent and heterogeneous renewal times by generalising Lorden's inequality through a generalised renewal intensity. It develops a parallel (successful) coupling framework to obtain explicit convergence-rate bounds to a stationary distribution in total variation without requiring explicit generators. The key contribution is a computable bound on the overshoot and an associated convergence-rate estimate for generalised renewal processes, enabling ergodicity results in reliability models and Markov-modulated settings. The approach offers a flexible, efficient alternative for analyzing complex networks and reliability systems where traditional matrix-analytic methods are impractical.

Abstract

We present a generalisation of Lorden's inequality tailored for stochastic systems with dependent and non-identically distributed renewal times modelled as generalised renewal processes. This framework introduces the new concept of a generalised intensity, enabling the analysis of systems where classical renewal theory is inapplicable. The proposed approach is applied to a variety of complex systems, including reliability models, queueing networks and Markov-modulated processes, to establish ergodicity and derive convergence rate estimates. Unlike traditional methods relying on transition matrices or generators, our technique provides an alternative with some rigorous upper bounds, but computationally much more efficient than existent ones. This work offers a unified framework for analysing systems with dependencies and heterogeneities, bridging gaps in classical stochastic modelling.

Paper Structure

This paper contains 23 sections, 14 theorems, 100 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Objectives of the research
  3. Structure of the Paper
  4. Classical Lorden's Inequality for Regenerative Processes
  5. Lorden's Inequality for the Classical Renewal Process
  6. Convergence Rate Estimates for Regenerative Processes
  7. Coupling Method and the Fundamental Coupling Inequality
  8. "Successful coupling"
  9. Constructing Convergence Rate Estimates Using Lorden's Inequality
  10. Distribution of the Renewal Process as a Function of the Overshoot $B_t$
  11. Intensity of the Renewal Process
  12. Convergence Rate Estimate for the Overshoot Distribution in the Classical Renewal Process
  13. Parallel Coupling Construction for the Process $B_t$
  14. Generalised Lorden's Inequality
  15. Generalised Renewal Process
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\xi$ be a random variable with distribution function $F(s)$, where $F(s)$ is non-latticeNon-lattice distribution means that $\xi$ is not concentrated only on points of a lattice $\{a + nd : n \in \mathbb{Z}\}$, where $a \in \mathbb{R}$ and $d > 0$. This condition ensures the validity of the pro

Figures (8)

  • Figure 1: Regenerative process.
  • Figure 2: The embedded renewal process within a regenerative process.
  • Figure 3: $B_t$ -- overshoot (backward recurrence time) at the fixed time $t$, $W_t$ -- undershoot (forward recurrence time) at the fixed time $t$; $t_i$ -- moments of "events".
  • Figure 4: Renewal process. Here $B_t=y$, and the probability of recovery is estimated on the interval $(t,t+\Delta)$.
  • Figure 5: Coupling construction of the recovery processes $B_t^{(1)}$ and $B_t^{(2)}$.
  • ...and 3 more figures

Theorems & Definitions (37)

  • Definition 1: Regenerative Process ross1996stochastic
  • Remark 1
  • Definition 2
  • Definition 3: Regeneration
  • Definition 4
  • Definition 5
  • Theorem 1: Lorden's Inequality Lorden1970
  • Theorem 2
  • Definition 6: Coupling Time
  • Theorem 3: Fundamental Coupling Inequality lindvall2002lecturesborovkov2022compound
  • ...and 27 more