Skip to main content
SpringerLink
Log in

The Large Deviation Principle for Finite-Dimensional Distributions of Multidimensional Renewal Processes

  • Published:
Siberian Advances in Mathematics Aims and scope Submit manuscript

Abstract

We study two types of multidimensional compound renewal processes (c.r.p.). We assume that the elements of the sequences that control the processes satisfy Cramér’s moment condition. Wide conditions are proposed under which the large deviation principle holds for finite-dimensional distributions of the processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. A. Borovkov, Probability Theory (Librokom, Moscow, 2009; Springer, London, 2013).

    Google Scholar 

  2. A. A. Borovkov, Asymptotic Analysis of Random Walks. Rapidly Decreasing Distributions of Increments (Fizmatlit, Moscow, 2013) [in Russian].

    MATH  Google Scholar 

  3. A. A. Borovkov and A. A. Mogul’skiĭ, “The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks,” Sibirsk. Mat. Zh. 37, 745 (1996) [Siberian Math. J. 37, 647 (1996)].

    Article  MathSciNet  Google Scholar 

  4. A. A. Borovkov and A. A. Mogul’skiĭ, “Large deviation principles for the finite-dimensional distributions of compound renewal processes,” Sibirsk. Mat. Zh. 56, 36 (2015) [Siberian Math. J. 56, 28 (2015)].

    Article  MathSciNet  Google Scholar 

  5. A. A. Borovkov and A. A. Mogul’skiĭ, “Integro-local limit theorems for compound renewal processes under Cramér’s condition. I,” Sibirsk. Mat. Zh. 59, 491 (2018) [Siberian Math. J. 59, 383 (2018)].

    Article  MathSciNet  Google Scholar 

  6. A. A. Borovkov and A. A. Mogul’skiĭ, “Integro-local limit theorems for compound renewal processes under Cramér’S condition. II,” Sibirsk. Mat. Zh. 59, 736 (2018) [Siberian Math. J. 59, 578 (2018)].

    Article  MathSciNet  Google Scholar 

  7. A. A. Borovkov, A. A. Mogul’skiĭ, and E. I. Prokopenko, “Properties of the deviation rate function and the asymptotics for the Laplace transform of the distribution of a compound renewal process,” Teor. Veroyatnost. i Primenen. 64 (4), 625 (2019) [Theory Probab. Appl., 64:4, 499 (2020)].

    Article  Google Scholar 

  8. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications (Springer, Berlin–Heidelberg, 2010).

    Book  Google Scholar 

  9. M. Kotulski, “Asymptotic distributions of continuous-time random walks: A probabilistic approch,” J. Statist. Phys. 81, 777 (1995).

    Article  Google Scholar 

  10. R. Lefevere, M. Mariani, and L. Zambotti, “Large deviations for renewal processes,” Stochastic Processes Appl. 121, 2243 (2011).

    Article  MathSciNet  Google Scholar 

  11. A. A. Mogul’skiĭ, “Local theorems for arithmetic compound renewal processes when Cramer’s condition holds,” Sib. Èlektron. Mat. Izv. 16, 21 (2019).

    Article  MathSciNet  Google Scholar 

  12. A. A. Mogul’skiĭ and E. I. Prokopenko, “Integro-local theorems for multidimensional compound renewal processes, when Cramer’s condition holds. I,” Sib. Èlektron. Mat. Izv. 15, 475 (2018).

    MathSciNet  MATH  Google Scholar 

  13. A. A. Mogul’skiĭ and E. I. Prokopenko, “Integro-local theorems for multidimensional compound renewal processes, when Cramer’s condition holds. II,” Sib. Èlektron. Mat. Izv. 15, 503 (2018).

    MathSciNet  MATH  Google Scholar 

  14. A. A. Mogul’skiĭ and E. I. Prokopenko, “Integro-local theorems for multidimensional compound renewal processes, when Cramer’s condition holds. III,” Sib. Èlektron. Mat. Izv. 15, 528 (2018).

    MathSciNet  MATH  Google Scholar 

  15. A. A. Mogul’skiĭ and E. I. Prokopenko, “Local theorems for arithmetic multidimensional compound renewal processes under Cramér’s condition,” Mat. Tr. 22 (2), 106 (2019) [Sib. Adv. Math. 30, 284 (2020)].

    Google Scholar 

  16. A. A. Mogul’skiĭ and E. I. Prokopenko, “The rate function and the fundamental function for multidimensional compound renewal process,” Sib. Èlektron. Mat. Izv. 16, 1449 (2019).

    Article  MathSciNet  Google Scholar 

  17. A. A. Mogul’skiĭ and E. I. Prokopenko, “Large deviation principle for multidimensional first compound renewal processes in the phase space,” Sib. Èlektron. Mat. Izv. 16, 1464 (2019).

    Article  MathSciNet  Google Scholar 

  18. A. A. Mogul’skiĭ and E. I. Prokopenko, “Large deviation principle for multidimensional second compound renewal processes in the phase space,” Sib. Èlektron. Mat. Izv. 16, 1478 (2019).

    Article  MathSciNet  Google Scholar 

  19. B. Tsirelson, “From uniform renewal theorem to uniform large and moderate deviations for renewal-reward processes,” Electron. Commun. Probab. 18, Paper No. 52, 13 p. (2013).

  20. M. Zamparo, Large Deviations in Discrete-Time Renewal Theory, https://arXiv.org/abs/1903.03537 (2019).

Download references

Funding

The work was supported by the Russian Science Foundation (grant no. 18-11-00129).

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    A. A. Mogul’skiĭ & E. I. Prokopenko

  2. ESSEC Business School, CREAR Risk Research Center, Cergy-Pontoise, 95021, France

    E. I. Prokopenko

Authors
  1. A. A. Mogul’skiĭ
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. I. Prokopenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. A. Mogul’skiĭ or E. I. Prokopenko.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mogul’skiĭ, A.A., Prokopenko, E.I. The Large Deviation Principle for Finite-Dimensional Distributions of Multidimensional Renewal Processes. Sib. Adv. Math. 31, 188–208 (2021). https://doi.org/10.1134/S1055134421030032

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1055134421030032

Keywords

Search

Navigation