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

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