By the converse of the strong law of large numbers, we know that, given a sequence of i.i.d random variables $X_1,X_2,\dots$ such that $\mathbb{P}(X_1 \ge 0)=1$ and $\mathbb{E}X_1= \infty$, then I have $$ S_N:=\frac{1}{N}\sum_{i=1}^N X_i \longrightarrow \infty \quad \mathbb{P}\textit{-a.s}. $$

I suppose that, just as in the case of the strong law of large numbers (without further assumptions on the moments of such random variables), we don't have a a-priori bound for the speed of divergence. My question is: given the law $\mathbb{P}_X$ of $X_1$, is there a well defined deterministic and diverging sequence $a_N=a_N(\mathbb{P}_X)$ and two positive but finite random variables $c$ and $C$ such that $$ \mathbb{P}\Big(\limsup_{N\to \infty} [S_N \ge C\cdot a_N] \Big)=0 $$ and $$ \mathbb{P}\Big(\limsup_{N\to \infty} [S_N \le c\cdot a_N] \Big)=0. $$

Is it possible to get bounds for $a_N$ in terms of the law $\mathbb{P}_X$?