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

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