If we assume that the $X_i$ are iid in $L^p$ for some $p\in ]1,2[$, then we have:

$$ a.e. \ \ {1 \over n} \sum_{k=0}^{n−1} X_k=E(X_0)+o(n^{1/p−1})$$

This follows from the Kolmogorov three series theorem. This is done in the book of Durrett, [probability, theory and examples](https://www.math.duke.edu/~rtd/PTE/PTE4_1.pdf), theorem 2.5.8.

Note that if you are not interested by the exact exponent, then the standard quick 
$L^2$ proof gives you such an estimate.