Convergence en loi et lois du logarithme itéré pour les vecteurs gaussiens

In this paper we prove a Strassen version of the law of the iterated logarithm for some sequences of weakly asymptotically independant Banach space valued gaussian random variables which converge in distribution, and we prove that the central limit theorem implies the functional form of the law of the iterated logarithm for the partial sums of certain Banach space valued gaussian sequences.Furthermore we give conditions for the convergence in distribution of sequences of gaussian random variables and gaussian stochastic processes, and these conditions permit us to prove that our results generalize in the gaussian case all similar results known to the authors at present.