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According to Hoeffding Inequality, if $X_1,\ldots,X_n$ are independent random variables with $\mathbb{P}(X_i \in [a_i,b_i]) = 1 \; \forall i = 1,\ldots,n$ then $$\mathbb{P}(\bar{X_n} - \mathbb{E}[\bar{X_n}] \geq t) \leqslant \exp\left(-\frac{2t^2n^2}{\sum_{i=1}^{n}(b_i - a_i)^2}\right) \quad \forall t > 0.$$ Does there exist any such bound for the probability of the event $\{\bar{X_n} - c\mathbb{E}[\bar{X_n}] \geqslant t\}$ for any constant $c$. Hoeffding Inequality is just a special case of this event when $c = 1$.

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You can just apply Hoeffding's Inequality, replacing $t$ with $t + (c - 1)\mathbb{E}[\bar{X_n}]$. –  Andrew Uzzell Dec 6 '12 at 21:16

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To bound this event, you can just apply Hoeffding's Inequality as usual, replacing $t$ with $t + (c - 1)\mathbb{E}[\bar{X_n}]$.

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