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Functional Limit Theorems for Compound Renewal Processes

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Abstract

We generalize Anscombe’s Theorem to the case of stochastic processes converging to a continuous random process. As applications, we find a simple proof of an invariance principle for compound renewal processes (CRPs) in the case of finite variance of the elements of the control sequence. We find conditions, close to minimal ones, of the weak convergence of CRPs in the metric space D with metrics of two types to stable processes in the case of infinite variance. They turn out narrower than the conditions for convergence of a distribution in this space.

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References

  1. Anscombe F. J., “Large–sample theory of sequential estimation,” Math. Proc. Cambridge Philos. Soc., vol. 48, No. 4, 600–607 (1952).

    Article  MathSciNet  MATH  Google Scholar 

  2. Gut A., Stopped Random Walks. Limit Theorems and Applications, Springer, New York (2009).

    Book  MATH  Google Scholar 

  3. Borovkov A. A., “On the distribution of the first passage time of a random walk to an arbitrary remote boundary,” Theory Probab. Appl., vol. 61, No. 2, 235–254 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  4. Billingsley P., Convergence of Probability Measures. Second edition, John Wiley & Sons, New York (1999).

    Book  MATH  Google Scholar 

  5. Borovkov A. A., Probability Theory, Gordon and Breach, Abingdon (1998).

    MATH  Google Scholar 

  6. Steinebach J., “Invariance principles for renewal processes when only moments of low order exist,” J. Multivariate Anal., vol. 26, No. 2, 169–183 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  7. Csörgö M., Horváth L., and Steinebach J., “Invariance principles for renewal processes,” Ann. Probab., vol. 15, No. 4, 1441–1460 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  8. Frolov A. N., “Limit theorems for increments of compound renewal processes,” J. Math. Sci. (New York), vol. 152, No. 6, 944–957 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  9. Borovkov A. A., “On the rate of convergence in the invariance principle for generalized renewal processes,” Theory Probab. Appl., vol. 27, No. 3, 461–471 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  10. Csörgö M., Deheuvels P., and Horváth L., “An approximation of stopped sums with applications in queueing theory,” Adv. Appl. Probab., vol. 19, No. 3, 674–690 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  11. Billingsley P. P., Convergence of Probability Measures [Russian translation], Nauka, Moscow (1977).

    Google Scholar 

  12. Borovkov A. A., Asymptotic Methods in Queueing Theory, John Wiley and Sons, Chichester etc. (1984).

    MATH  Google Scholar 

  13. Gikhman I. I. and Skorokhod A. V., The Theory of Random Processes. Vol. 1 [Russian], Nauka, Moscow (1971).

    MATH  Google Scholar 

  14. Smirnov N. V., “Limit distributions for the terms of a variational series,” Trudy Mat. Inst. Steklov., vol. 25, 3–60 (1949).

    MathSciNet  Google Scholar 

  15. Gnedenko B. V. and Sherif A., “Limit theorems for extreme terms of a variational series,” Dokl. Akad. Nauk SSSR, vol. 270, No. 3, 523–525 (1983).

    MathSciNet  MATH  Google Scholar 

  16. Borovkov A. A., “The convergence of distributions of functionals on stochastic processes,” Russian Math. Surveys, vol. 27, No. 1, 1–42 (1972).

    Article  MathSciNet  Google Scholar 

  17. Skorokhod A. V., “Limit theorems for stochastic processes,” Theory Probab. Appl., vol. 1, No. 3, 261–290 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  18. Gnedenko B. V. and Kolmogorov A. N., Limit Distributions for Sums of Independent Random Variables [Russian], Gostekhizdat, Moscow (1949).

    Google Scholar 

  19. Borovkov A. A. and Mogulskii A. A., “Large deviation principles for random walk trajectories,” Theory Probab. Appl., I: vol. 56, No. 4, 538–561 (2012); II: vol. 57, No. 1, 1–2 (2013); III: vol. 58, No. 1, 25–37 (2014).

    Article  MATH  Google Scholar 

  20. Borovkov A. A., Asymptotic Analysis of Random Walks: Rapidly Decreasing Jumps [Russian], Fizmatlit, Moscow (2013).

    Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia

    A. A. Borovkov

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  1. A. A. Borovkov
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Correspondence to A. A. Borovkov.

Additional information

The author was partially supported by the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (Program No. I.1.3, Project 0314-2016-0008) and the Russian Foundation for Basic Research (Grant 18-01-00101a).

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Translated from Sibirskii Matematicheskii Zhurnal, vol. 60, no. 1, pp. 37–54, January–February, 2019; DOI: 10.17377/smzh.2019.60.104.

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Borovkov, A.A. Functional Limit Theorems for Compound Renewal Processes. Sib Math J 60, 27–40 (2019). https://doi.org/10.1134/S003744661901004X

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  • DOI: https://doi.org/10.1134/S003744661901004X

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