Let
$$A_n = \sum_{k=1}^{n} {\bf x}_k{\bf x}_k^T $$
where ${\bf x}_k$ is a $p \times 1$ vector. Suppose that there is a $N \in \mathbb N$ such that the family of matrices $\{ A_n \}$ is uniformly positive definite for all $n > N$. Let $\{w_k\}$ be scalars uniformly bounded from above and away from zero. Is it true that we have
$$B_n = \sum_{k=1}^{n} w_k {\bf x}_k{\bf x}_k^T$$
also uniformly positive definite for $n > N$?
I have seen this result invoked in a paper but I've never seen it proven.