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July 1999 Renewal Theory for Embedded Regenerative Sets
Jean Bertoin
Ann. Probab. 27(3): 1523-1535 (July 1999). DOI: 10.1214/aop/1022677457


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.


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Jean Bertoin. "Renewal Theory for Embedded Regenerative Sets." Ann. Probab. 27 (3) 1523 - 1535, July 1999.


Published: July 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0961.60082
MathSciNet: MR1733158
Digital Object Identifier: 10.1214/aop/1022677457

Primary: 60K05

Keywords: Dirichlet distribution , multivariate renewal theory , Regenerative set

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 3 • July 1999
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