Let $X \in R^n$ be a random vector such that

$$P(|X_i| > \epsilon) > e^{-\epsilon^2}$$

What is a tight bound on

$$P(\sum_{i=1}^n |X_i| > \epsilon)$$

and on

$$P(\max_{1\le i\le n} |X_i| > \epsilon)?$$

The $X_i$ can be arbitrarily dependent. The best I can get for both bounds is $n e^{-\epsilon^2}$ (using the union bound).