Bounded law of the iterated logarithm for sums of independent random vectors normalized by matrices

Abstract

Let $(X_n,n\geq 1)$ be a sequence of independent centered random vectors in $\mathbf R^d$ with finite moments of order $p\in (2,3]$ and let $(A_n,n\geq 1)$ be a sequence of $m\times d$ matrices. We find explicit conditions under which \[ \limsup _{n\to \infty } c_n \left \|A_n\sum _{i=1}^n X_i\right \|<\infty \] almost surely, where $(c_n,n\geq 1)$ is some sequence of positive numbers.