# Large Deviations: Exponential decay in normed spaces

Let $(X_1,X_2,\cdots)$ be a sequence of independent and identically distributed random variables taking values in some general normed space $(V,||\cdot||)$. Denote $\mu=E[X_1]$ and $S_n=\frac{1}{n}[X_1+\cdots+X_n]$.

This question is about the tail probability $Pr[||S_n-\mu||\geq\epsilon]$ for some $\epsilon > 0$ and I am especially concerned with the question if the tail probability decays to exponentially fast in $n$.

Is $||X_i||\leq\tau$, $\tau \in (0,\infty)$ sufficient for Hoeffdings's Inequallity to hold with $$Pr[||S_n-\mu||\geq\epsilon]\leq \exp\left(-n^2\epsilon^2c \right),$$ where $c>0$ is some finite constant?