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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

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

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 $1-n e^{-\epsilon^2}$ (using the union bound).

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).

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

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

and on

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

The $X_i$ can be arbitrarily dependent.

A simpler question, which I believe can help to answer the question above, is the following: if $X_1, X_2\ge 0$ are two possibly dependent random variables such that $P(X_i > \epsilon) \le e^{-\epsilon^2}$, what is the best bound on $P(\max\left{X_1, X_2\right}> \epsilon)$?

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

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

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 $1-n e^{-\epsilon^2}$ (using the union bound).

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

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

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

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

and on

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

The $X_i$ can be arbitrarily dependent.

A simpler question, which I believe can help to answer the question above, is the following: if $X_1, X_2\ge 0$ are two possibly dependent random variables such that $P(X_i > \epsilon) \le e^{-\epsilon^2}$, what is the best bound on $P(\max\left{X_1, X_2\right}> \epsilon)$?