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Limit theorems for increments of compound renewal processes

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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.

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References

  1. P. Erdős and A. Rényi, “On a new law of large numbers,” J. Anal. Math., 23, 103–111 (1970).

    Article  Google Scholar 

  2. M. Csörgő and P. Révész, Strong Approximations in Probability and Statistics, Akadémiai Kiadó, Budapest (1981).

    Google Scholar 

  3. A. N. Frolov, “On asymptotic behavior of increments of sums of independent random variables,” Dokl. RAN, 272, 596–599 (2000).

    Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  5. A. N. Frolov, “Limit theorems for increments of sums of independent random variables,” Teor. Veroyatn. Primen., 48, 104–121 (2003).

    Google Scholar 

  6. A. N. Frolov, “Converses to the Csörgő-Révész laws,” Statist. Probab. Lett., 72, 113–123 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. N. Frolov, “Strong limit theorems for increments of renewal processes,” Zap. Nauchn. Semin. POMI, 298, 208–225 (2003).

    Google Scholar 

  8. A. N. Frolov, “On asymptotic behavior of increments of random fields,” Zap. Nauchn. Semin. POMI, 298, 191–207 (2003).

    Google Scholar 

  9. A. N. Frolov, “Universal limit theorems for increments of processes with independent increments,” Teor. Veroyatn. Primen., 49, 601–609 (2004).

    Google Scholar 

  10. A. N. Frolov, “Strong limit theorems for increments of sums of independent random variables,” Zap. Nauchn. Semin. POMI, 311, 260–285 (2004).

    MATH  Google Scholar 

  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. P. Embrechts and C. Klüppelberg, “Some aspects of insurance mathematics,” Theory Probab. Appl., 30, 374–416 (1993).

    Google Scholar 

  13. E. Seneta, Regularly Varying Functions [Russian translation], Moscow (1985).

  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).

  15. J. Galambos, Asymptotic Theory of Extreme Order Statistics [Russian translation], Moscow (1984).

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Authors and Affiliations

  1. St.Petersburg State University, St.Petersburg, Russia

    A. N. Frolov

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  1. A. N. Frolov
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Correspondence to A. N. Frolov.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 259–283.

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Frolov, A.N. Limit theorems for increments of compound renewal processes. J Math Sci 152, 944–957 (2008). https://doi.org/10.1007/s10958-008-9113-4

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  • DOI: https://doi.org/10.1007/s10958-008-9113-4

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