Having no answer so far, I asked this on MO now.

My question is motivated by this question and this question, where the first was aimed for giving a one sided tail bound for maximum of subgaussians, and the second one was for two sided tail bounds for gaussians. I'm also motivated by questions like this one.

Let $\{X_1 \dots X_n\}$ be $n$ iid, subgaussian random variables so that $P[|X_i| \ge t] \le 2 exp (- \frac{ct^2}{ ||X_1||_{\psi_2}^2 }) \forall i, ||*||_{\psi_2}$ denoting the Orlicz norm. I'm looking for concentration inequalities for :

$$ X_{max} := max_{1 \le i \le n} X_i, \hspace{1mm} X_{min} := min_{1 \le i \le n} X_i $$

So to be more precise, I'm looking for tail bound functions $\alpha(t), \beta(t)$ of the form:

$$ P[ | X_{max} - \mathbb{E}X_{max} | \ge t ] \le \alpha(t), P[ | X_{min} - \mathbb{E}X_{min} | \ge t ] \le \beta(t) $$

where the following are decreasing functions of $t$:

$$ 0 \le \alpha(t), \beta(t) \to 0, t \to \infty.$$



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