Let $(X_i)_{i\geq 1}$ be an i.i.d. sequence of mean zero random variables, $S_n:= X_1+\cdots + X_n$ and $V_n^2:=X_1^2+\cdots +X_n^2$. We consider four sequences of partial sums processes: the broken lines with vertices at the points $(k/n,S_k/V_n)$ or $(V_k^2/V_n^2,S_k/V_n)$ and the corresponding random step functions. We prove that each of them converges weakly in $C[0,1]$ or $D[0,1]$ to the Brownian motion \emph{if and only if} $X_1$ belongs to the \emph{domain of attraction of the normal distribution}. These results contrast with the classical Donsker Prohorov invariance principles where the N.S.C. for such convergences is $\E X_1^2 < \infty$.
Publié le : 2001-07-05
Classification:
self-normalization,
invariance principle,
domain of attraction,
functional central limit theorem,
AMS 60F05, 60B05, 60G17, 60E10,
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
@article{hal-00834538,
author = {Ra\v ckauskas, Alfredas and Suquet, Charles},
title = {Convergence of self-normalized partial sums processes in C[0,1] and D[0,1]},
journal = {HAL},
volume = {2001},
number = {0},
year = {2001},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00834538}
}
Račkauskas, Alfredas; Suquet, Charles. Convergence of self-normalized partial sums processes in C[0,1] and D[0,1]. HAL, Tome 2001 (2001) no. 0, . http://gdmltest.u-ga.fr/item/hal-00834538/