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 $E(X \mid Y) = E(Y \mid X) = 0$.
Publié le : 1983-08-14
Classification:
Sharp bounds on moments,
sum of two i.i.d. random variables,
martingales,
60E15,
60G50,
60G42
@article{1176993521,
author = {Cox, David C. and Kemperman, J. H. B.},
title = {Sharp Bounds on the Absolute Moments of a Sum of Two I.I.D. Random Variables},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 765-771},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993521}
}
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., Tome 11 (1983) no. 4, pp. 765-771. http://gdmltest.u-ga.fr/item/1176993521/