Siberian Mathematical Journal

, Volume 60, Issue 1, pp 27–40 | Cite as

Functional Limit Theorems for Compound Renewal Processes

  • A. A. BorovkovEmail author


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.


Anscombe’s theorem functional limit theorems compound renewal processes invariance principle convergence to a stable process 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anscombe F. J., “Large–sample theory of sequential estimation,” Math. Proc. Cambridge Philos. Soc., vol. 48, No. 4, 600–607 (1952).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gut A., Stopped Random Walks. Limit Theorems and Applications, Springer, New York (2009).CrossRefzbMATHGoogle Scholar
  3. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Billingsley P., Convergence of Probability Measures. Second edition, John Wiley & Sons, New York (1999).CrossRefzbMATHGoogle Scholar
  5. 5.
    Borovkov A. A., Probability Theory, Gordon and Breach, Abingdon (1998).zbMATHGoogle Scholar
  6. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Csörgö M., Horváth L., and Steinebach J., “Invariance principles for renewal processes,” Ann. Probab., vol. 15, No. 4, 1441–1460 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frolov A. N., “Limit theorems for increments of compound renewal processes,” J. Math. Sci. (New York), vol. 152, No. 6, 944–957 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Billingsley P. P., Convergence of Probability Measures [Russian translation], Nauka, Moscow (1977).Google Scholar
  12. 12.
    Borovkov A. A., Asymptotic Methods in Queueing Theory, John Wiley and Sons, Chichester etc. (1984).zbMATHGoogle Scholar
  13. 13.
    Gikhman I. I. and Skorokhod A. V., The Theory of Random Processes. Vol. 1 [Russian], Nauka, Moscow (1971).zbMATHGoogle Scholar
  14. 14.
    Smirnov N. V., “Limit distributions for the terms of a variational series,” Trudy Mat. Inst. Steklov., vol. 25, 3–60 (1949).MathSciNetGoogle Scholar
  15. 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).MathSciNetzbMATHGoogle Scholar
  16. 16.
    Borovkov A. A., “The convergence of distributions of functionals on stochastic processes,” Russian Math. Surveys, vol. 27, No. 1, 1–42 (1972).MathSciNetCrossRefGoogle Scholar
  17. 17.
    Skorokhod A. V., “Limit theorems for stochastic processes,” Theory Probab. Appl., vol. 1, No. 3, 261–290 (1956).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gnedenko B. V. and Kolmogorov A. N., Limit Distributions for Sums of Independent Random Variables [Russian], Gostekhizdat, Moscow (1949).Google Scholar
  19. 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).CrossRefzbMATHGoogle Scholar
  20. 20.
    Borovkov A. A., Asymptotic Analysis of Random Walks: Rapidly Decreasing Jumps [Russian], Fizmatlit, Moscow (2013).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia

Personalised recommendations