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

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

What is a tight bound on 

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

and on

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