negative correlation can improve concentration

Stronger concentration inequalities can sometimes be given for random variables which are negatively correlated.

The intuition is as follows: consider two random variables $X_1=X$ and $X_2=-X$. Then $\text{Cov}(X_1,X_2)=-{\sigma }^2<0$ where ${\sigma }^2=\mathbb{V}(X_1)$. Applying, say Chebyshev’s inequality gives

$$\mathbb{P}(|X_1+X_2|\ge t)\le \frac{\mathbb{V}(X_1+X_2)}{t^2}=\frac{{\sigma }^2}{t^2}.$$

But this is quite weak since, of course, $X_1$ and $X_2$ are perfectly concentrated around 0.

Negative correlation can be incorporated in the Chernoff method (see eg here). Garbe and Vondrak study the concentration of Lipschitz functions of negatively correlated rvs.