Let $X^0_1, \ldots, X^0_N$ be integer-valued random variables and let $a_1, \ldots, a_N$ be (fixed) nonzero vectors. We introduce the notion of coarseness of a discrete distribution and obtain upper bounds on the coarseness of the distribution of $S = \Sigma X^0_ia_i$ by comparison with the case $a_i \equiv a$. The bounds derived are seen to be tight and to apply for example when $S$ is formed (a) from independent summands or (b) by using any of a large family of sampling schemes. We show how such bounds can easily and efficiently substitute for use of Berry-Esseen theorems and other analytical methods in applications.
Publié le : 1988-10-14
Classification:
Random sums,
coarseness,
Berry-Esseen theorems,
top $K$ set,
top $K$ probability,
sampling,
stochastic monotonicity,
log concavity,
60E15,
60G50
@article{1176991589,
author = {Fill, James Allen},
title = {Bounds on the Coarseness of Random Sums},
journal = {Ann. Probab.},
volume = {16},
number = {4},
year = {1988},
pages = { 1644-1664},
language = {en},
url = {http://dml.mathdoc.fr/item/1176991589}
}
Fill, James Allen. Bounds on the Coarseness of Random Sums. Ann. Probab., Tome 16 (1988) no. 4, pp. 1644-1664. http://gdmltest.u-ga.fr/item/1176991589/