The object of this paper is to prove a central limit theorem in (separable) Hilbert space using a method based on the so called découpage de Lévy, the Lindeberg proof for the Gaussian case and an elementary proof of Poisson convergence for the direct part, and on elementary probabilistic inequalities for the converse. In particular, characteristic functions are only used in unicity questions. Several results of Varadhan (1962) can be obtained either directly as corollaries of the main theorem or with the same methods.
@article{urn:eudml:doc:38833,
title = {On the central limit theorem in Hilbert space.},
journal = {Stochastica},
volume = {4},
year = {1980},
pages = {43-71},
zbl = {0432.60011},
mrnumber = {MR0573725},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:38833}
}
Giné, Evarist; León, José R. On the central limit theorem in Hilbert space.. Stochastica, Tome 4 (1980) pp. 43-71. http://gdmltest.u-ga.fr/item/urn:eudml:doc:38833/