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Limit theorems for increments of compound renewal processes

  • A. N. Frolov
Article
  • 32 Downloads

Abstract

We derive universal strong limit theorems for increments of compound renewal processes which unify the strong law of large numbers, Erdős-Rényi law, Csörgő-Révész law, and law of the iterated logarithm for such processes. New results are obtained under various moment assumptions on distributions of random variables generating the process. In particular, we study the case of distributions from domains of attraction of the normal law and completely asymmetric stable laws with index α ∈ (1, 2). Bibliography: 15 titles.

Keywords

Russia Limit Theorem Renewal Process Strong Limit Iterate Logarithm 
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References

  1. 1.
    P. Erdős and A. Rényi, “On a new law of large numbers,” J. Anal. Math., 23, 103–111 (1970).CrossRefGoogle Scholar
  2. 2.
    M. Csörgő and P. Révész, Strong Approximations in Probability and Statistics, Akadémiai Kiadó, Budapest (1981).Google Scholar
  3. 3.
    A. N. Frolov, “On asymptotic behavior of increments of sums of independent random variables,” Dokl. RAN, 272, 596–599 (2000).Google Scholar
  4. 4.
    A. N. Frolov, “On one-sided strong laws for increments of sums of i.i.d. random variables,” Studia Sci. Math. Hungar., 39, 333–359 (2002).MATHMathSciNetGoogle Scholar
  5. 5.
    A. N. Frolov, “Limit theorems for increments of sums of independent random variables,” Teor. Veroyatn. Primen., 48, 104–121 (2003).Google Scholar
  6. 6.
    A. N. Frolov, “Converses to the Csörgő-Révész laws,” Statist. Probab. Lett., 72, 113–123 (2005).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. N. Frolov, “Strong limit theorems for increments of renewal processes,” Zap. Nauchn. Semin. POMI, 298, 208–225 (2003).Google Scholar
  8. 8.
    A. N. Frolov, “On asymptotic behavior of increments of random fields,” Zap. Nauchn. Semin. POMI, 298, 191–207 (2003).Google Scholar
  9. 9.
    A. N. Frolov, “Universal limit theorems for increments of processes with independent increments,” Teor. Veroyatn. Primen., 49, 601–609 (2004).Google Scholar
  10. 10.
    A. N. Frolov, “Strong limit theorems for increments of sums of independent random variables,” Zap. Nauchn. Semin. POMI, 311, 260–285 (2004).MATHGoogle Scholar
  11. 11.
    A. N. Frolov, “On the law of the iterated logarithm for sums of independent random variables,” Zap. Nauchn. Semin. POMI, 330, 174–186 (2004).Google Scholar
  12. 12.
    P. Embrechts and C. Klüppelberg, “Some aspects of insurance mathematics,” Theory Probab. Appl., 30, 374–416 (1993).Google Scholar
  13. 13.
    E. Seneta, Regularly Varying Functions [Russian translation], Moscow (1985).Google Scholar
  14. 14.
    A. N. Frolov, Limit theorems for increments of sums of independent random variables and stochastic processes, Dr. Sci. Thesis, St.Petersburg State Univ. (2004).Google Scholar
  15. 15.
    J. Galambos, Asymptotic Theory of Extreme Order Statistics [Russian translation], Moscow (1984).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.St.Petersburg State UniversitySt.PetersburgRussia

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