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

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

$\limsup_{n\to\infty} c_n \left\Vert A_n\sum_{i=1}^n X_i\right\Vert<\infty$

almost surely, where $(c_n,n\geq 1)$ is some sequence of positive numbers.