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An Inequality for Tail Probabilities of Martingales with Bounded Differences

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Abstract

Let M n =X1+...+Xn be a martingale with bounded differences Xm=Mm-Mm-1 such that ℙ{|Xm|≤ σ m}=1 with some nonnegative σm. Write σ2= σ 21 + ... +σ 2n . We prove the inequalities ℙ{M n≥x}≤c(1-Φ(x/σ)), ℙ{M n ⩾ x}⩽ 1- c(1-Φ (-x/σ)) with a constant \(c \leqslant 1(1 - \Phi (\sqrt 3 )) \leqslant 25\). The result yields sharp inequalities in some models related to the measure concentration phenomena.

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REFERENCES

  1. V. Bentkus, Large deviations in Banach spaces, Probab. Theory Appl., 31(4), 710–716 (1986).

    Google Scholar 

  2. V. Bentkus, An inequality for large deviation probabilities of sums of bounded i.i.d.r.v., Lith. Math. J., 41(2), 112–119 (2001).

    Google Scholar 

  3. V. Bentkus and M. van Zuijlen, Upper confidence bounds for mean, Ann. Statist. (to appear).

  4. W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58, 13–30 (1963).

    Google Scholar 

  5. M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, in: Séminaire de Probabilite´s, XXXIII, Springer, Berlin (1999), pp. 120–216.

    Google Scholar 

  6. C. McDiarmid, On the method of bounded differences, London Math. Soc. Lecture Note Ser., 141, 148–188 (1989).

    Google Scholar 

  7. V. Paulauskas and A. Ra?kauskas, Approximation Theory in the Central Limit Theorem. Exact Results in Banach Spaces, Kluwer, Dordrecht (1989).

    Google Scholar 

  8. V. V. Petrov, Sums of Independent Random Variables, Springer, New York (1975).

    Google Scholar 

  9. I. Pinelis, Extremal probabilistic problems and Hotelling's T 2 test under a symmetry assumption, Ann. Statist., 22(4), 357–368 (1994).

    Google Scholar 

  10. G. R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics, Wiley, New York (1986).

    Google Scholar 

  11. M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes E´tudes Sci. Publ. Math., 81, 73–205 (1995).

    Google Scholar 

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  1. Vilnius Institute of Mathematics and Informatics, Akademijos 4, LT-2600, Vilnius, Lithuania

    V. Bentkus

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  1. V. Bentkus
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Bentkus, V. An Inequality for Tail Probabilities of Martingales with Bounded Differences. Lithuanian Mathematical Journal 42, 255–261 (2002). https://doi.org/10.1023/A:1020269808826

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  • DOI: https://doi.org/10.1023/A:1020269808826

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