Convergence Rate of Sample Average Estimator

You may have a look at the large deviations theory.

In this case, when the distribution is regular enough, the Cramer theorem states:

$$ \frac 1n\log P\left(\frac 1n [X_1 + \dots + X_n] > \mu + \epsilon\right) \to -\left[\sup_{t\in \Bbb R} (\mu + \epsilon)t - \log Ee^{tX} \right] $$

The condition being in that case that $ Ee^{tX} $ exists. So the good rate of convergence is $$ P\left(\left|\frac 1n [X_1 + \dots + X_n] - \mu\right| \ge \epsilon\right) \simeq e^{-\alpha n} $$with $\alpha $ given by the right hand side of the preceding equality.