Given only a bound of the form $P(|X_i| < c_1) < 1 - c_2$ for some fixed small $c_1, c_2 > 0$, does there exist $c>0$ such that
$$P\left(\sum_{i=1}^n |X_i| < \varepsilon n\right) = O(e^{-cn})$$ for sufficiently small $\epsilon$? Here the $X_i$ are iid.
The probability that all of the $|X_i|$ are less than $\varepsilon$ is $(1-c_2)^n$, but I do not see how to get an exponentially small bound on the average being less than $\varepsilon$.
This comes from Exercise 2 of Terence Tao's blog post 254A Notes 7: The Least Singular Value