The best constant in the Topchii-Vatutin inequality for martingales

Gerold Alsmeyer and Uwe Rösler

Statistics & Probability Letters, 2003, vol. 65, issue 3, 199-206

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.

Keywords: Martingale; Topchii-Vatutin; inequality; Convex; function; Choquet; representation (search for similar items in EconPapers)
Date: 2003
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Citations: View citations in EconPapers (2)

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