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Update: I was a bit imprecise with my centering argument: of course we cannot assume non-negativity of the RVs themselves (we cannot "center" unbounded RVs like Gaussians), but we can make their expectations non-negative wlog. . Then we have that $\mathbb{P}(\hat{\mu}_k- \mu_k \geq \epsilon) \leq \mathbb{P}(\hat{\mu}_k \geq \epsilon)\leq \exp(- \dots)$.