Fastest way to converge to mean

Question

Given a sequence of i.i.d random variables $\{X_k\}_{k=1}^N$, from the central limit theorem, we know that $$ \left(\frac{1}{N}\sum_{k=1}^N X_k \right)- \mathbb{E}(X_1) = \mathcal{o}_p\left(1\right) $$ Is there any way to obtain an estimator for mean with a faster rate of convergence in probability?

Answer 1

If $X_1$ has a finite variance, then the law of the iterated logarithms states that the random variable $$ M=\sup_{n\geqslant 3}\frac 1{\sqrt{n \log \log n}}\left\lvert \sum_{k=1}^n(X_k-\mathbb E(X_k))\right\rvert $$is almost surely finite. Therefore, $$ \left\lvert\frac 1N\sum_{k=1}^NX_k-\mathbb E(X_1)\right\rvert=\frac 1N \left\lvert\sum_{k=1}^N(X_k-E[X_k])\right\rvert\leqslant \frac{\sqrt{\log \log N}}{\sqrt N}M. $$