Let $X_1, X_2,\cdots$ be a sequence of independent identically distributed random vectors and $S_n = X_1 + \cdots + X_n$. Necessary and sufficient conditions are given for there to exist matrices $B_n$ and vectors $\gamma_n$ such that $\{B_n(S_n - \gamma_n)\}$ is stochastically compact, i.e., $\{B_n(S_n - \gamma_n)\}$ is tight and no subsequential limit is degenerate. When this condition holds we are able to obtain precise estimates on the distribution of $S_n$. These results are then specialized to the case where $X_1$ is in the generalized domain of attraction of an operator stable law and a local limit theorem is proved which generalizes the classical local limit theorem where the normalization is done by scalars.
Publié le : 1986-01-14
Classification:
Matrix normalization,
stochastic compactness,
tightness,
probability estimates,
local limit theorem,
generalized domain of attraction,
60F05
@article{1176992624,
author = {Griffin, Philip S.},
title = {Matrix Normalized Sums of Independent Identically Distributed Random Vectors},
journal = {Ann. Probab.},
volume = {14},
number = {4},
year = {1986},
pages = { 224-246},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992624}
}
Griffin, Philip S. Matrix Normalized Sums of Independent Identically Distributed Random Vectors. Ann. Probab., Tome 14 (1986) no. 4, pp. 224-246. http://gdmltest.u-ga.fr/item/1176992624/