On a class of martingale inequalities

David C. Cox and J. H. B. Kemperman

Journal of Multivariate Analysis, 1983, vol. 13, issue 2, 328-352

Abstract: Let Z = {Z0, Z1, Z2,...} be a martingale, with difference sequence X0 = Z0, Xi = Zi - Zi - 1, i >= 1. The principal purpose of this paper is to prove that the best constant in the inequality [lambda]P(supi Xi >= [lambda]) 0, is C = (log 2)-1. If Z is finite of length n, it is proved that the best constant is Cn = [n(21/n - 1)]-1. The analogous best constant Cn(z) when Z0 [reverse not equivalent] z is also determined. For these finite cases, examples of martingales attaining equality are constructed. The results follow from an explicit determination of the quantity Gn(z, E) = supz P(maxi=1,...,n Xi >= 1), the supremum being taken over all martingales Z with Z0 [reverse not equivalent] z and EZn = E. The expression for Gn(z,E) is derived by induction, using methods from the theory of moments.

Keywords: Martingale; inequalities; best; constants (search for similar items in EconPapers)
Date: 1983
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