Consider zero-mean sub-exponential random variables $\{X_1,...,X_n\}$ (not necessarily independent) with parameters $(\nu, \alpha)$. That is, $$\mathbb E[\exp\{\lambda X\}] \le \exp\{\nu^2\lambda^2/2\}, \qquad ^\forall |\lambda| < 1/\alpha.$$ Check whether $\max_{1\le i \le n}X_i$ is sub-exponential, and if so, its tail probability .
This is what I want to study.
If each $X_i$ is independent, I guess I can easily see $Y := \max_{1\le i \le n}X_i$ is sub-exponential using the following property:
$$\begin{align} \mathbb E[\exp\{\lambda Y\}] & \le \mathbb E[\exp\{\lambda \sum_{i=1}^nX_i\}]\\ & = \mathbb E \left(\exp\{\lambda X\}\right)^n \\ & \le \exp\{n\nu^2\lambda^2/2\} \end{align}$$
But, I can't use this directly when $X_i$ are not independent. So, I need some help about the properties of $Y := \max_{1\le i \le n}X_i$ w.r.t. sub-exponentials.
Any comment about this question would be grateful. Thank you.
Union bounding, $P(e^{tY}\geq u)\leq nP(e^{tX}\geq u)$. Namely, $Y$ is sub-exponential.