# Stationary coupling method for renewal process in continuous time (application to strong bounds for the convergence rate of the distribution of the regenerative process)

## TL;DR

This paper introduces a stationary coupling method for continuous-time renewal processes, providing improved bounds on the convergence rate of regenerative process distributions, especially under heavy tail conditions.

## Contribution

It develops a novel stationary coupling technique that enhances classical polynomial convergence bounds for regenerative processes with heavy tails.

## Key findings

- Improved convergence bounds for regenerative processes.
- Demonstrates effectiveness of stationary coupling in total variation metrics.
- Extends previous results with stronger convergence rate estimates.

## Abstract

We propose a new modification of the coupling method for renewal process in continuous time. We call this modification "the stationary coupling method", and construct it primarily to obtain the bounds for convergence rate of the distribution of the regenerative processes in the total variation metrics. At the same time this modification of the coupling method demonstrates an improvement of the classical result of polynomial convergence rate of the distribution of the regenerative process -- in the case of a heavy tail. This paper is an extension of my preprint "On strong bounds of rate of convergence for regenerative processes" - arXiv:1608.02243