Intersections and limits of regenerative sets

  • P. J. Fitzsimmons
  • Bert Fristedt
  • B. Maisonneuve


Regenerative subsets of ℝ constitute an analog of classical renewal processes. Limits and intersections of independent regenerative sets are discussed. These ideas are related to the usual quantities associated with subordinators.


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  1. 1.
    Billingsley, P.: Convergence of probability measures. New York: Wiley 1968Google Scholar
  2. 2.
    Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York-London: Academic Press 1968Google Scholar
  3. 3.
    Fristedt, B.E.: Sample functions of stochastic processes with stationary independent increments. In Advances in Probability, 3, New York: Marcel Dekker 1974Google Scholar
  4. 4.
    Fristedt, B.E.: The central limit problem for, infinite products of, and Levy processes of renewal sequences. Z. Wahrscheinlichkeitstheor. Verw. Geb.58, 479–507 (1981)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hawkes, J.: Intersections of Markov random sets. Z. Wahrscheinlichkeitstheor. Verw. Geb.37, 243–251 (1977)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Hoffmann-Jørgensen, J.: Markov sets. Math. Scand.24, 145–166 (1969)MathSciNetGoogle Scholar
  7. 7.
    Kendall, D.G.: Delphic semigroups, infinitely divisible regenerative phenomena and the arithmetic ofp-functions. Z. Wahrscheinlichkeitstheor. Verw. Geb.9, 163–195 (1968)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Kendall, D.G.: Foundations of a theory of random sets. In Stochastic Geometry. pp. 322–376. London-New York: Wiley 1974Google Scholar
  9. 9.
    Kingman, J.F.C.. Regenerative phenomena. London-New York: Wiley 1972Google Scholar
  10. 10.
    Krylov, N.V., Yushkevich, A.A.: Markov random sets. Trans. Mosc. Math. Soc.13, 127–153 (1965)MATHGoogle Scholar
  11. 11.
    Maisonneuve, B.: Ensembles régénératifs, temps locaux et subordinateurs. In Sèminaire de Probabilités V, Lecture Notes in Mathematics191. Berlin-Heidelberg-New York: Springer 1971Google Scholar
  12. 12.
    Maisonneuve, B.: Systemes régénératifs. Astérisque15, Société Mathématique de France 1974Google Scholar
  13. 13.
    Maisonneuve, B.: Changement de temps d'un processus markovien additif. In Séminaire de Probabilités XI. Lecture Notes in Mathematics581. Berlin-Heidelberg-New York: Springer 1977Google Scholar
  14. 14.
    Maisonneuve, B.: Ensembles régénératifs de la droite. Z. Wahrscheinlichkeitstheor. Verw. Geb.63, 501–510 (1983)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Matheron, G.: Random sets and integral geometry. London-New York. Wiley 1975Google Scholar
  16. 16.
    Meyer, P.A.: Ensembles régéneratifs, d'après Hoffmann-Jørgensen. In Séminaire de Probabilités IV. Lecture Notes in Mathematics124. Berlin-Heidelberg-New York: Springer 1970Google Scholar
  17. 17.
    Taksar, M.I.: Regenerative sets on real line. In Séminaire de Probabilités XIV. Lecture Notes in Mathematics784. Berlin-Heidelberg-New York: Springer 1980Google Scholar
  18. 18.
    Kesten, H.: Hitting probabilities of single points for processes with stationary independent increments. Mem. Am. Math. Soc.93, (1969)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • P. J. Fitzsimmons
    • 1
  • Bert Fristedt
    • 2
  • B. Maisonneuve
    • 3
  1. 1.Department of Mathematical SciencesUniversity of AkronAkronUSA
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  3. 3.I.M.S.S.Grenoble CedexFrance

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