On ε-coupling and piecewise constant processes

We show that if a stochastic process (Z_s)_s∈[0,∞) on a general state space with piecewise constant paths having finitely many jumps in finite intervals admits ε-coupling to a stationary process (Z_s*)_s∈[0,∞) for each ε > 0 then Z_t tends in total variation to Z0* as t → ∞. This result is applied in renewal theory to the total life process, to processes regenerative in the wide sense (regeneration as in Harris chains), and to the queue GI/GI/k with traffic intensity strictly between 0 and 1 but without assuming that the system ever empties.