Skip to main content

Large deviation principles for sums of random vectors and the corresponding renewal functions in the inhomogeneous case

Abstract

Under the inhomogeneous case wemean the case when one or several (arbitrarily many) inhomogeneous summands are added to the sum of independent identically distributed vectors. We find necessary and sufficient conditions under which the large deviation principles for such sums and the corresponding renewal functions have the same form that in the homogeneous case.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    S. Asmussen and H. Albrecher, Ruin Probabilities (World Scientific Publishing Co., Hackensack, NJ, 2010).

    MATH  Google Scholar 

  2. 2.

    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) [SiberianMath. J. 37, 647 (1996)].

    Google Scholar 

  3. 3.

    A. A. Borovkov and A. A. Mogul’skiĭ, “Integro-local limit theorems including large deviations for sums of random vectors. II,” Teor. Veroyatnost. i Primenen. 45, 5 (2000) [Theory Probab. Appl. 45, 3 (2001)].

    Article  Google Scholar 

  4. 4.

    A. A. Borovkov and A. A. Mogul’skiĭ, “On large deviation principles in metric spaces,” Sibirsk. Mat. Zh. 51, 1251 (2010) [SiberianMath. J. 51, 989 (2010)].

    Google Scholar 

  5. 5.

    A. A. Borovkov and A. A. Mogul’skiĭ, “Chebyshev-type exponential inequalities for sums of random vectors and for trajectories of random walks,” Teor. Veroyatnost. i Primenen. 56, 3 (2011) [Theory Probab. Appl. 56, 21 (2012)].

    Article  Google Scholar 

  6. 6.

    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)].

    Google Scholar 

  7. 7.

    D. R. Cox and W. L. Smith, Renewal Theory (Sovetskoe Radio, Moscow, 1967) [in Russian].

    MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. A. Borovkov.

Additional information

Original Russian Text © A.A. Borovkov, A.A. Mogul’skiĭ, 2014, published in Matematicheskie Trudy, 2014, Vol. 17, No. 2, pp. 84–101.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Borovkov, A.A., Mogul’skiĭ, A.A. Large deviation principles for sums of random vectors and the corresponding renewal functions in the inhomogeneous case. Sib. Adv. Math. 25, 255–267 (2015). https://doi.org/10.3103/S1055134415040033

Download citation

Keywords

  • large deviation principles
  • inhomogeneous sum of random vectors
  • renewal function
  • deviation rate function
  • second deviation rate function