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
$$P(\max_{1\le i\le n} |X_i| \ge \epsilon)?$$
concentration inequality for averages of dependent random variables
gappy3000
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