Universal normalization problem for the Gál–Koksma lemma

Let $(X_n)_{n\ge 1}$ be a sequence of random variables and let $(a_n)_{n\ge 1}$ be nonnegative numbers, with $A_n:=\sum_{k=1}^n a_k\to\infty$. Assume that there is a constant $C<\infty$ such that, for every $1\le m\le n$, $\mathbb{E}\left|\sum_{k=m}^n X_k\right|^2\le C\sum_{k=m}^n a_k$. Determine exactly which non-decreasing functions $\psi$ have the universal property that every such admissible sequence satisfies $\sum_{k=1}^n X_k=O\!\left(\sqrt{A_n}\,\psi(A_n)\right)$ almost surely. The characterization is required to hold uniformly over the entire class described by the consecutive-block second-moment hypothesis.