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On coupling of renewal processes and its applications

The coupling, a probabilistic device dating back to Doeblin [1], has enjoyed a revival in recent years; see Griffeath [2] for a survey and references. Its power in the study of Markov chain transitions and renewal sequence asymptotics has been demonstrated in, among other papers, a thorough investigation by Pitman [3]. The purpose of this contribution is to show how some new ideas on coupling of renewal processes render a new proof of a key lemma of Pitman's, as well as extensions and improvements of some results due to him and Stone and Wainger [4].