We obtain the almost sure approximation of the partial sums of random variables with values in a separable Hilbert space and satisfying a strong mixing condition by a suitable Brownian motion. This is achieved by a modification of the proof of a similar result by Kuelbs and Philipp (1980) on $\phi$-mixing Banach space valued random variables. As by-products we get almost sure invariance principles for sums of absolutely regular sequences of random variables with values in a Banach space and necessary and sufficient conditions for the almost sure invariance principle for sums of independent, identically distributed random variables.
Publié le : 1982-08-14
Classification:
Almost sure invariances principles,
mixing and absolutely regular sequences of random variables,
Hilbert space,
Banach space,
Brownian motion,
60B12
@article{1176993777,
author = {Dehling, Herold and Philipp, Walter},
title = {Almost Sure Invariance Principles for Weakly Dependent Vector-Valued Random Variables},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 689-701},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993777}
}
Dehling, Herold; Philipp, Walter. Almost Sure Invariance Principles for Weakly Dependent Vector-Valued Random Variables. Ann. Probab., Tome 10 (1982) no. 4, pp. 689-701. http://gdmltest.u-ga.fr/item/1176993777/