Let $X_1, X_2, \cdots, X_n$ be nonnegative bounded random variables either independent or negatively associated (stronger condition than negatively correlated). Let $\mathbb{E}\left [\frac{1}{n} \sum_{i=1}^n X_i \right ] = \mu$.
Then by Chebyshev, we have approximately,
$$\mathbb{P}\left( \frac{1}{n}\sum_{i=1}^n X_i - \mu > \Delta \right) \leq \frac{1}{n \Delta^2},$$
or by Chernoff-Hoeffding, we get approximately
$$\mathbb{P}\left( \frac{1}{n}\sum_{i=1}^n X_i - \mu > \Delta \right) \leq \exp \left( - n \Delta^2 \right).$$
There are other concentration inequalities too but in each case to get concentration (i.e., get decaying probability), we need to choose $\Delta = \frac{1}{n^{\frac{1}{2} - \epsilon}}.$ for some $\epsilon > 0$.
That $\epsilon > 0$ is really messing up a result I am trying to prove. Are there any extra regularity conditions I can impose or perhaps use other concentration inequalities which could help me get concentration for deviations equal to $1/\sqrt{n}$?