We give strong bounds for the rate of convergence of the
regenerative process distribution to the stationary distribution in the
total variation metric. For this aim we propose a new modification of
the coupling method which we call stationary coupling method. Use of
this stationary coupling method improves the classic results about the
convergence rate of the distribution of the regenerative process in the case
of a heavy tail. Also this method can be applied for obtaining the bounds
of the rate of the convergence for the queueing regenerative processes.