Let $X$ be a real valued random variable with probability distribution function $F(x)$ and characteristic function $c(t)$. Let $F_n(x)$ be the $n$th empirical distribution function associated with $X$ and $c_n(t)$ the characteristic function of $F_n(x)$. Necessary and sufficient conditions are obtained for the weak convergence of $\sqrt{n}\lbrack c_n(t) - c(t)\rbrack$ on the space of continuous complex valued functions on $\lbrack -1/2, 1/2\rbrack$.
Publié le : 1981-04-14
Classification:
Empirical characteristic function,
empirical distribution theorem,
central limit theorem on $C(\lbrack - 1/2, 1/2 \rbrack)$,
subgaussian processes,
60B10,
60F05
@article{1176994461,
author = {Marcus, Michael B.},
title = {Weak Convergence of the Empirical Characteristic Function},
journal = {Ann. Probab.},
volume = {9},
number = {6},
year = {1981},
pages = { 194-201},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994461}
}
Marcus, Michael B. Weak Convergence of the Empirical Characteristic Function. Ann. Probab., Tome 9 (1981) no. 6, pp. 194-201. http://gdmltest.u-ga.fr/item/1176994461/