Negatively correlated random variables Chebyshev bound?

Question

It is quite well known that the Chernoff bound applies to negatively correlated binary random variables (see e.g. Theorem 1.16 here). Does there exist a reference for Chebyshev-type bound for negatively correlated binary variables?

Answer 1

Why do you need a reference, instead of just deriving it or stating it as a fact? If the r.v.'s $(X_k)_{1\leq k \leq n}$ are negatively correlated (and have finite variance), then you have $$ \operatorname{Var}\sum_{k=1}^n X_k \leq \sum_{k=1}^n \operatorname{Var} X_k $$ (since all covariances are non-positive: expand the variance on the LHS); and therefore you can use Chebyshev's inequality directly: $$ \forall a>0,\qquad \mathbb{P}\left\{\left\lvert\sum_{k=1}^n X_k - \sum_{k=1}^n \mathbb{E}X_k\right\rvert > a\right\}\leq \frac{\operatorname{Var}\sum_{k=1}^n X_k}{a^2} \leq \frac{\sum_{k=1}^n \operatorname{Var}X_k}{a^2} \tag{$\dagger$} $$