This paper concerns moment and tail probability inequalities and the
strong law of large numbers for $U$-statistics with nonnegative or
symmetrized kernels and their multisample and decoupled versions. Sub-Bernoulli
functions are used to obtain the moment and tail probability inequalities,
which are then used to obtain necessary and sufficient conditions for the
almost sure convergence to zero of normalized $U$-statistics with
nonnegative or completely symmetrized kernels, without further regularity
conditions on the kernel or the distribution of the population, for normalizing
constants satisfying a simple condition. Moments of $U$-statistics are
bounded from above and below by that of maxima of certain kernels, up to
scaling constants. The multisample and decoupled versions of these results are
also considered.
Publié le : 1999-01-14
Classification:
Sub-Bernoulli function,
strong law of large numbers,
moment inequality,
exponential inequality,
tail probability,
$U$-statistics,
60F15,
60G50
@article{1022677268,
author = {Zhang, Cun-Hui},
title = {Sub-Bernoulli Functions, Moment Inequalities and Strong Laws for
Nonnegative and symmetrized U-Statistics},
journal = {Ann. Probab.},
volume = {27},
number = {1},
year = {1999},
pages = { 432-453},
language = {en},
url = {http://dml.mathdoc.fr/item/1022677268}
}
Zhang, Cun-Hui. Sub-Bernoulli Functions, Moment Inequalities and Strong Laws for
Nonnegative and symmetrized U-Statistics. Ann. Probab., Tome 27 (1999) no. 1, pp. 432-453. http://gdmltest.u-ga.fr/item/1022677268/