Let (Xn)n>j be a sequence of independent centered random vectors in Rd We give conditions under which the sequence Sn = JZJLj Xi normalized by a matricial sequence (Hn) satisfies a compact law of the iterated logarithm. As an application of this result, we obtain the I/O 1/2 compact law of the iterated logarithm for Bn Sn and for An Sn, where Bn is the covariance matrix of Sn, and where An is the diagonal matrix whose jth diagonal term is the jth diagonal term of Bn; the eigenvalues of Bn may go to infinity with different rates, but their iterated logarithms have to be equivalent.
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