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 is a standard exercice. Use the fact that if $Y_i$ is centered, independent such that $\sum Var(Y_i)$ is convergent, then the series $\sum Y_i$ converges a.e. Then take $Y_i = X_i/i^\alpha$ and conclude with the Kronecker lemma. This is a standard lemma used in the proof of 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.