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 $\mathbb{P}\_X$?



On the speed of divergence of the converse of the Strong law of large numbers
Such a sequence $a\_n$ does not exist even for a well studied example like returns to the origin of simple random walk in one dimension. If $X\_i$ denotes the number of steps from the $i-1$ time the walk returned to the origin to the $i$'th time, then $X\_i$ are i.i.d. and their sum $S\_n$ is the number of steps until the $n$'th return time to the origin. In [1] $S\_n$ is denoted by $\rho\_n$ and we will follow this. Taking $\epsilon=1$ in Theorem 11.6 on page 119 in [1] we find that
$$\rho\_n>n^2 \log (n)$$ infinitely often almost surely, yet
$$\rho\_n<\frac{n^2}{\log \log n}$$
infinitely often almost surely. More precise information is in Theorem 11.5 there. To exclude "wild" sequences $a\_n$, just separate cases, using Theorem 9.11 in [1] that yields for all $x \in (0,\infty)$,
$$\lim\_{k \to \infty} P(\rho\_k <k^2 x) = f(x) , $$
where $f(x) \in (0,1)$ is explicitly given.


Case 1: If $a\_n \le n^2$ infinitely often, then for any constant $C \in (0,\infty)$ we will have $P(\limsup [\rho\_n>Ca\_n]) \ge 1-f(C) $ by Fatou's lemma so
$P(\limsup [\rho\_n>Ca\_n]) =1 $ by the Hewitt-Savage zero-one law.


Case 2: If $a\_n \le n^2$ finitely often, then a similar argument shows that $P(\limsup [\rho\_n \le c a\_n]) =1 $ for all $c \in (0,\infty)$.


The situation is more extreme for return times of two dimensional simple random walk, for which see Theorem 20.5 page 218 in [1].


[1] P. Revesz, Random Walk in Random and Non-Random Environments, World Scientific
Publ., Second edition (2005). <https://www.worldscientific.com/worldscibooks/10.1142/5847>


