On the range of a regenerative sequence (Q1062682)

For a given countable partition of the range of a regenerative sequence \(\{X_ k;k\geq 0\}\) let \(R_ n\) be the number of distinct sets from the partition visited by \(\{X_ k;k\leq n\}\). Under appropriate conditions the limiting behaviour of \(R_ n\) and of \(ER_ n\) is derived and illustrated by examples. The original set up is then broadened to cover the case of a Harris recurrent Markov chain generalizing a result of \textit{L. Chosid} and \textit{R. Isaac}, Ann. Probab. 6, 680-687 (1978; Zbl 0388.60064); Corrections, ibid. 8, 1000 (1980).