Skip to main content
Log in

Large Deviations for Sums of Random Vectors Attracted to Operator Semi-Stable Laws

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

Let \(\{X_i, i\ge 1\}\) be i.i.d. \(\mathbb {R}^d\)-valued random vectors attracted to operator semi-stable laws and write \(S_n=\sum _{i=1}^{n}X_i\). This paper investigates precise large deviations for both the partial sums \(S_n\) and the random sums \(S_{N(t)}\), where N(t) is a counting process independent of the sequence \(\{X_i, i\ge 1\}\). In particular, we show for all unit vectors \(\theta \) the asymptotics

$$\begin{aligned} {\mathbb P}(|\langle S_n,\theta \rangle |>x)\sim n{\mathbb P}(|\langle X,\theta \rangle |>x) \end{aligned}$$

which holds uniformly for x-region \([\gamma _n, \infty )\), where \(\langle \cdot , \cdot \rangle \) is the standard inner product on \(\mathbb {R}^d\) and \(\{\gamma _n\}\) is some monotone sequence of positive numbers. As applications, the precise large deviations for random sums of real-valued random variables with regularly varying tails and \(\mathbb {R}^d\)-valued random vectors with weakly negatively associated occurrences are proposed. The obtained results improve some related classical ones.

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

Access this article

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. Chorny, V.: Operator semistable distributions on \(\mathbb{R}^{d}\). Theory Probab. Appl. 57, 703–705 (1986)

    MATH  Google Scholar 

  2. Chover, J.: A law of the iterated logarithm for stable summands. Proc. Am. Math. Soc. 17, 441–443 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cline, D.B.H., Hsing, T.: Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Preprint. Texas A&M University (1991)

  4. Denuit, M., Lef\(\grave{e}\)vre, C., Utev, S.: Measuring the impact of dependence between claims occurrences. Insur. Math. Econ. 30(1), 1–19 (2002)

  5. Deuschel, J.-D., Stroock, D.W.: Large Deviations. Academic Press, Boston (1989)

    MATH  Google Scholar 

  6. Doeblin, W.: Sur l’ensemble de puissance d’une loi de probabilit\(\acute{e}\)cole Norm Ser. 63(3), 317–350 (1947)

  7. Ebrahimi, N., Ghosh, M.: Multivariate negative dependence. Commun. Stat. A 10, 307–337 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fuk, D.H., Nagaev, S.V.: Probability inequalities for sums of independent random variables. Theory Probab. Appl. 16, 660–675 (1971)

    Article  MATH  Google Scholar 

  9. Gnedenko, B.V., Korolev, V.Y.: Random Summation. Limit Theorems and Applications. CRC Press, Boca Raton (1996)

    MATH  Google Scholar 

  10. Han, K.-H.: A weak negative orthant dependence. Commun. Korean Math. Soc. 12(3), 755–768 (1997)

    MathSciNet  MATH  Google Scholar 

  11. Heyde, C.C.: A contribution to the theory of large deviations for sums of independent random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 7, 303–308 (1967a)

    Article  MathSciNet  MATH  Google Scholar 

  12. Heyde, C.C.: On large deviation problems for sums of random variables which are not attracted to the normal law. Ann. Math. Stat. 38, 1575–1578 (1967b)

    Article  MathSciNet  MATH  Google Scholar 

  13. Heyde, C.C.: On large deviation probabilities in the case of attraction to a nonnormal stable law. Sankhy\(\bar{a}\) 30, 253–258 (1968)

  14. Hudson, W.N., Veeh, J.A., Weiner, D.C.: Moments of distributions attracted to operator stable laws. J. Multivar. Anal. 24, 1–10 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jajte, R.: Semistable probability measures on \(\mathbb{R}^N\). Studia Math. 61, 29–39 (1977)

    MathSciNet  MATH  Google Scholar 

  16. Joag-Dev, K., Proschan, F.: Negative association of random variables with applications. Ann. Stat. 11, 286–295 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jurek, Z.J., Mason, J.D.: Operator-Limit Distribution in Probability Theory. A Wiley-Interscience Publication, Wiley, New York (1993)

  18. Jurek, Z.J., Wu, L.-M.: Large deviations for some Poisson random integrals. Lecture Notes in Mathematics, Springer, Berlin 1927, 185–197 (2000)

  19. Klüppelberg, C., Mikosch, T.: Large deviations of heavy-tailed random sums with applications in insurance and finance. J. Appl. Prob. 34, 293–308 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lin, Z., Bai, Z.-D.: Probability Inequalities. Science Press, Beijing (2010)

    MATH  Google Scholar 

  21. Meerschaert, M.M.: Spectral decomposition for generalized domains of attraction. Ann. Probab. 19, 875–892 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Meerschaert, M.M., Scheffler, H.P.: Limit Distributions for Sums of Independent Random Vectors. Heavy Tails in Theory and Practice. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (2001)

  23. Mikosch, T., Nagaev, A.V.: Large deviations of heavy-tailed sums with applications in insurance. Extremes 1(1), 81–110 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nagaev, A.V.: Integral limit theorems for large deviations when Cramers condition is not fulfilled I, II. Theory Prob. Appl. 14(51–64), 193–208 (1969a)

    Article  Google Scholar 

  25. Nagaev, A.V.: Limit theorems for large deviations when Cramers conditions are violated. Fiz-Mat. Nauk. 7, 17–22 (1969b). (in Russian)

    Google Scholar 

  26. Nagaev, S.V.: Large deviations for sums of independent random variables. In: Transactions of the Sixth Prague Conference on Information Theory, Random Processes and Statistical Decision Functions. Academic, Prague, 657–674 (1973)

  27. Nagaev, S.V.: Large deviations of sums of independent random variables. Ann. Probab. 7, 745–789 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ng, K.W., Tang, Q., Yan, J.-A., Yang, H.: Precise large deviations for sums of random variables with consistently varying tails. J. Appl. Probab. 41(1), 93–107 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Phuc, H.: Domains of operator semi-attraction of operator semistable probability measures. Acta Math. Vietnam 34, 257–267 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Rozovski, L.V.: Probabilities of large deviations on the whole axis. Theory. Probab. Appl. 38, 53–79 (1993)

    Article  MathSciNet  Google Scholar 

  31. Rüschendorf, L.: Weak association of random variables. J. Multivar. Anal. 11, 448–451 (1981)

    Article  MathSciNet  Google Scholar 

  32. Sato, K.I.: Lévy Processes and Infinitely Divisable Distributions. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  33. Scheffler, H.P.: Moments of measures attracted to operator semi-stable laws. Stat. Probab. Lett. 24, 187–192 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Scheffler, H.P.: A law of the iterated logarithm for heavy tailed random vectors. Probab. Theory Relat. Fields 116, 257–271 (2000a)

    Article  MathSciNet  MATH  Google Scholar 

  35. Scheffler, H.P.: Multivariate R-O varying measures part I: uniform bounds. Proc. Lond. Math. Soc. 81, 231–256 (2000b)

    Article  MATH  Google Scholar 

  36. Scheffler, H.P.: Multivariate R-O varying measures. Part II: individual bounds. J. Math. Sci. 111, 3868–3878 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  37. Shao, Q.M.: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theor. Probab. 13(2), 343–356 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Stout, W.F.: Almost Sure Convergence. Academic Press, New York (1974)

    MATH  Google Scholar 

  39. Tang, Q., Su, C., Jiang, T., Zhang, J.: Large deviations for heavy-tailed random sums in compound renewal model. Stat. Probab. Lett. 52(1), 91–100 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. Vinogradov, V.: Refined Large Deviation Limit Theorems. Longman, Harlow (1994)

    MATH  Google Scholar 

  41. Wang, W.: Invariance principles for generalized domains of semistable attraction. Stoch. Process. Appl. 124, 1–17 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author wishes to express his deep gratitude to the referees for their valuable comments on an earlier version which improved the quality of this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wensheng Wang.

Additional information

This research was supported by NSFC Grant 11071076 and NSF of Zhejiang Province Grant LY14A010025.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, W. Large Deviations for Sums of Random Vectors Attracted to Operator Semi-Stable Laws. J Theor Probab 30, 64–84 (2017). https://doi.org/10.1007/s10959-015-0645-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-015-0645-5

Keywords

Mathematics Subject Classification (2010)

Navigation