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Let $x_1, \ldots, x_n$ be $n$ i.i.d. samples of a bounded random variable $X \in [a, b]$. We know from the Hoeffding's inequality that :

$$\mathbb{P} \left( \left| \frac{1}{n} \sum_{i=1}^n x_n - \mathbb{E}[X] \right| \geq \varepsilon \right) \leq 2 \exp\left( - \frac{2 n \varepsilon^2}{(b-a)^2}\right)$$

Now let $x_1, \ldots, x_n$ be $n$ i.i.d. samples of a bounded random vector $X$. Do we have a similar bound for the following quantity ?

$$\mathbb{P} \left( \left\| \frac{1}{n} \sum_{i=1}^n x_n - \mathbb{E}[X] \right\|_1 \geq \varepsilon \right)$$

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