Backward limits and inhomogeneous regeneration

Consider a time-inhomogeneous regenerative process starting from regeneration at time s. It is shown (under regularity conditions on the regeneration times) that, as the starting time s tends backward to - ∞, the process tends in total variation in any fixed time-interval [t, ∞) to a two-sided limit process. Further, time-uniform rates of convergence are obtained, the analogue of the renewal measure is considered and a time-inhomogeneous key renewed theorem presented