Let $X_1, X_2, \ldots $ be independent random
variables with zero means and finite variances. It is well known
that a finite exponential moment assumption is necessary for a
Cramér-type large deviation result for the standardized partial
sums. In this paper, we show that a Cramér-type large deviation
theorem holds for self-normalized sums only under a finite
$(2+\delta)$th moment, $0< \delta \leq 1$. In particular, we show
$P(S_n /V_n \geq x)=\break
(1-\Phi(x)) (1+O(1) (1+x)^{2+\delta} /d_{n,\delta}^{2+\delta})$
for $0 \leq x \leq d_{n,\delta}$,\vspace{1pt} where $d_{n,\delta} =
(\sum_{i=1}^n EX_i^2)^{1/2}/(\sum_{i=1}^n
E|X_i|^{2+\delta})^{1/(2+\delta)}$ and $V_n= (\sum_{i=1}^n
X_i^2)^{1/2}$. Applications to the Studentized bootstrap and to
the self-normalized law of the iterated logarithm are discussed.
Publié le : 2003-10-14
Classification:
Large deviation,
moderate deviation,
nonuniform Berry--Esseen bound,
self-normalized sum,
t-statistic,
law of the iterated logarithm,
Studentized bootstrap,
60F10,
60F15,
60G50,
62F03
@article{1068646382,
author = {Jing, Bing-Yi and Shao, Qi-Man and Wang, Qiying},
title = {Self-normalized Cram\'er-type large deviations for independent random variables},
journal = {Ann. Probab.},
volume = {31},
number = {1},
year = {2003},
pages = { 2167-2215},
language = {en},
url = {http://dml.mathdoc.fr/item/1068646382}
}
Jing, Bing-Yi; Shao, Qi-Man; Wang, Qiying. Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab., Tome 31 (2003) no. 1, pp. 2167-2215. http://gdmltest.u-ga.fr/item/1068646382/