Let $x_1,..., x_n$ be independent identically distributed variables with means $M$ and variances $V$. Let $\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i.$
Then we can say that:
$\mathbb{P}(|\bar{x}-M|>\varepsilon)\leq\frac{V}{n\varepsilon^2}. $
Is this bound sharp for these conditions? If it is true can you hint me of an example that shows that it can't be tightened?
Thanks!