Let $S_n := \sum_{i=1}^n X_i$, where $X_i$ are mean $0$ variance $1$ i.i.d random variables. By Donsker's theorem, we know that $$ \frac{1}{N} \sum_{n=1}^N f \left( \frac{S_n}{\sqrt{n}} \right) \to \int_0^1 dt \ f (W(t))$$ weakly as $n \to \infty$, where $f \in C_b (\mathbb{R})$ is a continuous and bounded function, and $W(t)$ is a standard Brownian motion on $[0,1]$.
Is there some simple way of showing(if it is true) that $\frac{1}{N} \sum_{n=1}^N f \left( \frac{S_n}{\sqrt{n}} \right)$ does not converge almost surely to any random variable for some $f \in C_b(\mathbb{R})$? Usually, one can make use of some facts such as convergence in probability implying the existence of an almost surely convergent subsequence and thus arriving at some kind of situation in which one can apply Borel-Cantelli for a contradiction.
Similarily, if one wishes to show that $\frac{S_n}{\sqrt{n}}$ does not converge almost surely, it is enough to use, for instance, the law of the iterated logarithm to construct almost surely divergent subsequences.
The arithmetic mean is messing with my attempts, since results like recurrence of the random walk yield subsequences of $S_{n_k}$ rather than of the arithmetic mean.