Compact Law of the Iterated Logarithm for Matrix-Normalized Sums of Random Vectors

Let $(X_n)_{n\ge 1}$ be a sequence of independent centered random vectors in $R^d$. We give conditions under which the sequence $S_n=\sum_{i=1}^nX_i$ normalized by a matricial sequence $(H_n)$ satisfies a compact law of the iterated logarithm. As an application of this result, we obtain the compact law of the iterated logarithm for $B_n^{-1/2}S_n$ and for $\Delta_n^{-1/2}S_n$, where $B_n$ is the covariance matrix of $S_n$, and where $\Delta_n$ is the diagonal matrix whose $j$th diagonal term is the $j$th diagonal term of $B_n$; the eigenvalues of $B_n$ may go to infinity with different rates, but their iterated logarithms have to be equivalent.