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, 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.