The Annals of Probability

Renewal Theory for Embedded Regenerative Sets

Jean Bertoin

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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.

Article information

Source
Ann. Probab., Volume 27, Number 3 (1999), 1523-1535.

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677457

Digital Object Identifier
doi:10.1214/aop/1022677457

Mathematical Reviews number (MathSciNet)
MR1733158

Zentralblatt MATH identifier
0961.60082

Subjects
Primary: 60K05: Renewal theory

Keywords
Multivariate renewal theory regenerative set Dirichlet distribution

Citation

Bertoin, Jean. Renewal Theory for Embedded Regenerative Sets. Ann. Probab. 27 (1999), no. 3, 1523--1535. doi:10.1214/aop/1022677457. https://projecteuclid.org/euclid.aop/1022677457


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References

  • 1 BERTOIN, J. 1997. Regenerative embedding of Markov sets. Probab. Theory Related Fields 108 559 571.
  • 2 BERTOIN, J. 1999. Subordinators: Examples and Applications. Ecole d'ete de Probabilites de ´ ´ ´ St-Flour XXVII. Lecture Notes in Math. Springer, Berlin. To appear.
  • 3 BINGHAM, N. H., GOLDIE, C. M. and TEUGELS, J. L. 1987. Regular Variation. Cambridge Univ. Press.
  • 4 FELLER, W. E. 1968. An Introduction to Probability Theory and Its Applications, 1 3rd ed. Wiley, New York.
  • 5 FELLER, W. E. 1971. An Introduction to Probability Theory and Its Applications, 2, 2nd ed. Wiley, New York.
  • 6 FRISTEDT, B. E. 1996. Intersections and limits of regenerative sets. In Random DiscreteStructures D. Aldous and R. Pemantle, eds. 121 151. Springer, Berlin.
  • 7 HOGLUND, T. 1988. A multidimensional renewal theorem. Bull. Sci. Math. 112 111 138. ¨
  • 8 SPITZER, F. 1986. A multidimensional renewal theorem. In Probability, Statistical Mechanics, and Number Theory G. C. Rota, ed. 147 155. Academic Press, Orlando.
  • LABORATOIRE DE PROBABILITES, UMR 7599 ´ UNIVERSITE PIERRE ET MARIE CURIE ´ PARIS FRANCE F-75252 E-MAIL: jbe@ccr.jussieu.fr