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

Abstract

Let (Xn, n≥1) be a sequence of independent centered random vectors in Rd. We study the law of the iterated logarithm lim supn→∞(2 log log ‖Bn‖)−1/2 ‖B−1/2nSn‖=1 a.s., where Bn is the covariance matrix of Sn=∑ni=1Xi, n≥1. Application to matrix-normalized sums of independent random vectors is given.