Skip to main content
SpringerLink
Log in

Limit theorems for large deviations probabilities of certain quadratic forms

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract

Let (ξ k ,F k ) be a martingale difference sequence. The paper concerns the tail behavior of the quadratic formS n = ∑ n k=1 ∑ k−1 j=1 β k−j n χ k χ j , where β n asn→∞. The main conclusions aboutP}n −1 S n >x n }, wherex n →∞, asn→∞, are obtained using the tail behavior of a martingale with values in a certain Hilbert space.

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. V. Ventkus, On large deviations in Banach spaces,Theory Probab. Appl.,31, 710–716 (1986).

    Google Scholar 

  2. V. Bentkus and A. Račkauskas, On probabilities of large deviations in Banach spaces,Probab. Theory Related Fields,86, 131–154 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  3. E. Bolthausen, Exact convergence rates in some martingale central limit theorems,Ann. Probab.,10, 672–688 (1982).

    MATH  MathSciNet  Google Scholar 

  4. N. N. Chan and C. Z. Wei, Asymptotic inference for nearly nonstationaryAR(1) processes,Ann. Statist.,15, 1050–1063 (1987).

    MATH  MathSciNet  Google Scholar 

  5. D. D. Cox and I. Llatas, Maximum likelihood type estimation for nearly nonstationary autoregressive time series,Ann. Statist.,19, 1109–1128 (1991).

    MATH  MathSciNet  Google Scholar 

  6. I. G. Grama, The probabilities of large deviations for semimartingales,Stochanstics Stochastics Rep.,54, 1–20 (1995).

    MATH  MathSciNet  Google Scholar 

  7. P. Hitczenko, Best constants in martingale version of Rosenthal’s inequality,Ann. Probab.,18, 1656–1668 (1990).

    MATH  MathSciNet  Google Scholar 

  8. P. Hall and C. C. Heyde,Martingale Limit Theory and Its Application, Academic Press, New York (1980).

    Google Scholar 

  9. I. A. Ibragimov and Yu. V. Linnik,Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen (1971).

    Google Scholar 

  10. M. Ledoux and M. Talagrand,Probability in Banach Spaces, Springer-Verlag, Berlin-Heidelberg-New York (1991).

    Google Scholar 

  11. A. Račkauskas, On probabilities of large deviations for martingale,Liet. Mat. Rinkinys,30, 784–795 (1990).

    Google Scholar 

  12. A. Račkauskas, On probabilities of large deviations for martingales in Hilbert space, in:Proc. Seventh Japan-Russia Symp. Prob. Th. Math. Stat. (Tokyo, 1995), World Sc. Publ. Corp., Singapore (1996), pp. 423–437.

    Google Scholar 

  13. V. V. Petrov,Sums of Independent Random Variables, Springer, Berlin-Heidelberg-New York (1975).

    Google Scholar 

  14. L. Saulis and V. Statulevičius,Limit Theorems for Large Deviations, Kluwer, Dordrecht (1989).

    Google Scholar 

  15. V. V. Yurinsky, On large deviations asymptotics in Hilbert space. I,Theory Probab. Appl.,36, 99–114 (1991); II,Theory Probab. Appl.,36, 548–554 (1991); III,Theory Probab. Appl.,37, 261–268 (1992).

    Article  MathSciNet  Google Scholar 

  16. H. Walk, An invariance principle for the Robbins-Monroe process in a Hilbert space,Z. Wahrsch. verw. Geb.,39, 135–150 (1977).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Authors
  1. A. Račkauskas
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Vilnius University, Naugarduko 24; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 4, pp. 532–549, October–December, 1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Račkauskas, A. Limit theorems for large deviations probabilities of certain quadratic forms. Lith Math J 37, 402–415 (1997). https://doi.org/10.1007/BF02465581

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02465581

Keywords

Search

Navigation