The Annals of Probability

Sharp Bounds on the Absolute Moments of a Sum of Two I.I.D. Random Variables

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

Full-text: Open access

Abstract

We determine the exact optimal bounds $A_p$ and $B_p$ such that $A_p\lbrack E|X|^p + E|Y|^p\rbrack \leq E|X + Y|^p \leq B_p\lbrack E|X|^p + E|Y|^p\rbrack,$ $(p \geq 1)$, whenever $X, Y$ are i.i.d. random variables with mean zero. We give examples of random variables attaining equality or nearly so. Exactly the same bounds $A_p$ and $B_p$ are obtained in the more general case where it is only assumed that $E(X \mid Y) = E(Y \mid X) = 0$.

Article information

Source
Ann. Probab., Volume 11, Number 3 (1983), 765-771.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993521

Digital Object Identifier
doi:10.1214/aop/1176993521

Mathematical Reviews number (MathSciNet)
MR704563

Zentralblatt MATH identifier
0524.60019

JSTOR
links.jstor.org

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60G50: Sums of independent random variables; random walks 60G42: Martingales with discrete parameter

Keywords
Sharp bounds on moments sum of two i.i.d. random variables martingales

Citation

Cox, David C.; Kemperman, J. H. B. Sharp Bounds on the Absolute Moments of a Sum of Two I.I.D. Random Variables. Ann. Probab. 11 (1983), no. 3, 765--771. doi:10.1214/aop/1176993521. https://projecteuclid.org/euclid.aop/1176993521


Export citation