We prove general invariance principles for integral functions of the empirical process. As corollaries we derive probabilistic proofs of the sufficiency criteria for a distribution to belong to the domain of attraction of the normal and stable laws with index $0 < \alpha < 2$. In the process we obtain equivalent statements of these criteria in terms of the tail behaviour of the underlying quantile function. We also give a representation of any stable random variable with index $0 < \alpha < 2$ in terms of a linear combination of two independent and identically distributed Poisson integrals. The role of a fixed number of extreme terms is exactly determined.
Publié le : 1986-01-14
Classification:
Integral functionals,
empirical distribution function,
normal convergence criteria,
stable convergence criteria,
quantiles,
Poisson integrals,
60F17,
60F05,
60E07
@article{1176992618,
author = {Csorgo, Miklos and Csorgo, Sandor and Horvath, Lajos and Mason, David M.},
title = {Normal and Stable Convergence of Integral Functions of the Empirical Distribution Function},
journal = {Ann. Probab.},
volume = {14},
number = {4},
year = {1986},
pages = { 86-118},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992618}
}
Csorgo, Miklos; Csorgo, Sandor; Horvath, Lajos; Mason, David M. Normal and Stable Convergence of Integral Functions of the Empirical Distribution Function. Ann. Probab., Tome 14 (1986) no. 4, pp. 86-118. http://gdmltest.u-ga.fr/item/1176992618/