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The best constant in the Topchii-Vatutin inequality for martingales

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  • Alsmeyer, Gerold
  • Rösler, Uwe

Abstract

Consider the class of even convex functions with [phi](0)=0 and concave derivative on (0,[infinity]). Given any [phi]-integrable martingale (Mn)n[greater-or-equal, slanted]0 with increments , n[greater-or-equal, slanted]1, the Topchii-Vatutin inequality (Theory Probab. Appl. 42 (1997) 17) asserts thatwith C=4. It is proved here that the best constant in this inequality is C=2 for general [phi]-integrable martingales (Mn)n[greater-or-equal, slanted]0, and C=1 if (Mn)n[greater-or-equal, slanted]0 is further nonnegative or having symmetric conditional increment distributions.

Suggested Citation

  • Alsmeyer, Gerold & Rösler, Uwe, 2003. "The best constant in the Topchii-Vatutin inequality for martingales," Statistics & Probability Letters, Elsevier, vol. 65(3), pages 199-206, November.
    • Handle: RePEc:eee:stapro:v:65:y:2003:i:3:p:199-206
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    References listed on IDEAS

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    1. Alsmeyer, Gerold, 1996. "Nonnegativity of odd functional moments of positive random variables with decreasing density," Statistics & Probability Letters, Elsevier, vol. 26(1), pages 75-82, January.
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    Cited by:

    1. Alsmeyer, Gerold & Dyszewski, Piotr, 2017. "Thin tails of fixed points of the nonhomogeneous smoothing transform," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 3014-3041.
    2. G. Alsmeyer & U. Rösler, 2004. "On the Existence of φ-Moments of the Limit of a Normalized Supercritical Galton–Watson Process," Journal of Theoretical Probability, Springer, vol. 17(4), pages 905-928, October.

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