Let $W_n := \sum_{j=1}^n Z_j$ be a sum of independent
integer-valued random variables. In this paper, we derive an asymptotic
expansion for the probability $\mathbb{P}[W_n \in A]$ of an arbitrary subset $A
\in \mathbb{Z}$. Our approximation improves upon the classical expansions by
including an explicit, uniform error estimate, involving only easily computable
properties of the distributions of the $Z_j:$ an appropriate number of moments
and the total variation distance $d_{\mathrm{TV}}(\mathscr{L}(Z_j),
\mathscr{L}(Z_j + 1))$. The proofs are based on Stein’s method for
signed compound Poisson approximation.
Publié le : 2002-04-14
Classification:
compound Poisson,
Stein's method,
total variation distance,
Kolmogorov's problem,
60G50,
60F05,
62E20
@article{1023481001,
author = {Barbour, A. D. and \'Cekanavi\'cius, V.},
title = {Total variation asymptotics for sums of independent integer
random variables},
journal = {Ann. Probab.},
volume = {30},
number = {1},
year = {2002},
pages = { 509-545},
language = {en},
url = {http://dml.mathdoc.fr/item/1023481001}
}
Barbour, A. D.; Ćekanavićius, V. Total variation asymptotics for sums of independent integer
random variables. Ann. Probab., Tome 30 (2002) no. 1, pp. 509-545. http://gdmltest.u-ga.fr/item/1023481001/