A New Law of the Iterated Logarithm in Rd with Application to Matrix-Normalized Sums of Random Vectors

Let (X n , n>=1) be a sequence of independent centered random vectors in R d . We study the law of the iterated logarithm lim sup n (infinity)(2 log log ||B n ||)–1/2 ||B –1/2 n S n ||=1 a.s., where B n is the covariance matrix of S n =(sigma) n i=1 X i , n>=1. Application to matrix-normalized sums of independent random vectors is given.