Inequalities relating maximal moments to other measures of dispersion

P. C. Allaart

Statistica Neerlandica, 2000, vol. 54, issue 3, 366-373

Abstract: Let X, X1, ..., Xk be i.i.d. random variables, and for k∈ N let Dk(X) = E(X1 V ... V Xk+1) −EX be the kth centralized maximal moment. A sharp lower bound is given for D1(X) in terms of the Lévy concentration Ql(X) = supx∈ R P(X∈[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.

Date: 2000
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