Skip to main content
SpringerLink
Log in

On Probability and Moment Inequalities for Supermartingales and Martingales

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

The probability inequality for sum S n =∑ j=1 n X j is proved under the assumption that the sequence S k , k=\(\overline {1,n,}\), forms a supermartingale. This inequality is stated in terms of the tail probabilities P(X j >y) and conditional variances of the random variables X j , j=\(\overline {1,n,}\). The well-known Burkholder moment inequality is deduced as a simple consequence.

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. Burkholder, D. L.: Distribution function inequalities for martingales, Ann.Probab.1(1) (1973), 19–42.

    Google Scholar 

  2. Courbot, B.: Rates of convergence in the functional CLT for martingales, C.R.Acad.Sci.Paris 328(1999), 509–513.

    Google Scholar 

  3. De la Peña, V. H.: A general class of exponential inequalities for martingales and ratios, Ann. Probab.27(1) (1999), 537–554.

    Google Scholar 

  4. Dehling, H. and Utev, S. A.: An exponential inequality for martingales, Sib.Adv.Math.J.3(3) (1993), 197–203.

    Google Scholar 

  5. Dzhaparidze, H. and van Zanten, J. H.: On Bernstein-type inequalities for martingales, Stochastic Process.Appl.93(1) (2001), 109–118.

    Google Scholar 

  6. Fuk, D. Kh.: Some probabilistic inequalities for martingales, Siberian Math.J.14(1) (1973), 131–137.

    Google Scholar 

  7. Fuk, D. Kh. and Nagaev, S. V.: Some probability inequalities for sums of independent random variables, Theory Probab.Appl.16(4) (1971), 643–660.

    Google Scholar 

  8. Haeusler, E.: An exact rate of convergence in the functional central limit theorem for special martingale difference arrays, Z.Warsch.verw.Gebiete 65(4) (1984), 523–534.

    Google Scholar 

  9. Ibragimov, R. and Sharakhmetov, Sh.: On the sharp constant in Rosenthal inequality, Theor. Probab.Appl.42(2) (1997), 294–302.

    Google Scholar 

  10. Johnson, W. B., Schechtman, G. and Zinn, J.: Best constants in moment inequalities for linear combination of independent and exchangeable random variables, Ann.Probab.13(1985), 234–253.

    Google Scholar 

  11. Kubilius, K. and Mémin, J.: Inégalité exponentielle pour les martingales locales, C.R.Acad. Sci.Paris 319(1994), 733–737.

    Google Scholar 

  12. Lesign, E. and Volny, D.: Large deviations for martingales, Stochastic Process.Appl.66(1) (2001), 143–159.

    Google Scholar 

  13. Nagaev, S. V.: Large deviations of sums independent random variables, Ann.Probab.7(5) (1979), 745–789.

    Google Scholar 

  14. Nagaev, S. V.: Probability inequalities for sums of independent random variables taking values in a Banach space (in Russian), In: Limit Theorems of the Theory of Probabilities and Related Topics, Proc. Inst. Math. Siberian Branch USSR Acad. Sci. 1, 1982, pp. 159–167.

    Google Scholar 

  15. Nagaev, S. V.: Probability inequalities for sums of independent random variables with values in a Banach space, Siberian.Math.J.28(4) (1987), 652–664.

    Google Scholar 

  16. Nagaev, S. V.: Some refinements of probabilistic and moment inequalities, Theory Prob.Appl.42(4) (1997), 707–713.

    Google Scholar 

  17. Nagaev, S. V. and Pinelis, I. F.: On large deviations for sums of independent random variables taking values in a Banach space, In: Abstr.Comm.Second Vilnius Conf.Probab.Theory and Math.Stat., Vol. 2, Vilnius, 1977, pp. 66–67.

    Google Scholar 

  18. Peshkir, G. and Shiryaev, A. N.: The Khinchine inequalities and martingale extending of sphere of their action, Russian Math.Surveys 50(5) (1995), 849–1099.

    Google Scholar 

  19. Pinelis, I. F.: Optimum bounds for the distributions of martingales in Banach spaces, Ann. Probab.22(1994), 1679–1706.

    Google Scholar 

  20. Rosenthal, H. P.: On the subspaces of L p (p >2 spanned by sequences of independent random variables, Israel J.Math.8(3) (1970), 273–303.

    Google Scholar 

  21. Van de Geer, S.: Exponential inequalities for martingales with applications to maximum likelihood estimation for counting process, Ann.Statist.23(5) (1995), 1779–1801.

    Google Scholar 

  22. Volodin, N. A. and Morozova, L. N.: Some estimates of probabilities of large deviations for martingales and sums of random vectors (in Russian), In: Stochastic Processes and Mathematical Statistics,FAN, Tashkent, 1978, pp. 35–43.

    Google Scholar 

  23. Yurinskii, V. V.: Exponential inequalities for large deviations, Theor.Probab.Appl.19(1) (1974), 152–153.

    Google Scholar 

Download references

Authors
  1. S. V. Nagaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nagaev, S.V. On Probability and Moment Inequalities for Supermartingales and Martingales. Acta Applicandae Mathematicae 79, 35–46 (2003). https://doi.org/10.1023/A:1025814306357

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1025814306357

Search

Navigation