Summary: Inequalities relating maximal moments
to other measures of dispersion
Pieter C. Allaart
March 31, 2000
University of North Texas
Let X X1 : : : Xk be i.i.d. random variables, and for k 2 IN let Dk(X) =
E(X1 _ _ Xk+1) ; EX be the k-th centralized maximal moment. A sharp
lower bound is given for D1(X) in terms of the Levy concentration Ql(X) =
supx2IR P (X 2 x x + l]). This inequality, which is analogous to P. Levy's
concentration-variance inequality, illustrates the fact that maximal moments
are a gauge of how much spread out the underlying distribution is. It is also
shown that the centralized maximal moments are increased under convolution.
AMS 1990 subject classi cations: 60E15, 60G70.
Keywords and phrases: expected maximum of an i.i.d. random sample, Levy
concentration, measure of dispersion.
Author's address: Mathematics Department, University of North Texas, Den-
ton, TX 76203-1430, USA e-mail: email@example.com