Renewal Theory for Embedded Regenerative Sets

Abstract

We consider the age processes $A ^{(1)}\geq\cdots\geq A^{(n)}$ associated to a monotone sequence $\mathscr{R}^{(1)}\subseteq\cdots\subseteq\mathscr{R}^{(n)}$ of regenerative sets. We obtain limit theorems in distribution for (A_t^{(1)},\ldots, A_t^{(n)})$ and for $((1/t) A_t^{(1)},\ldots,(1/t)A_t^{(n)})$, which correspond to multivariate versions of the renewal theorem and of the Dynkin–Lamperti theorem, respectively. Dirichlet distributions play a key role in the latter.