I have $p$-dimensional independent vectors $\boldsymbol{X}_1,\ldots,\boldsymbol{X}_n.$ I have also condition: for some $\eta>0$:\begin{equation}\mathbb{E}\left(e^{tX_{ij}^2}\right)\leq K<\infty \quad\text{for}\quad |t|\leq\eta_j,\quad \forall i,j\end{equation}
I have also another condidion: for some $\gamma>1:$\begin{equation}\mathbb{E}\left(|X_{ij}|^{2(1+\gamma)}\right)\leq K\quad\forall i,j\end{equation}
My aim is to obtain an upper bound for $\mathbb{P}\left(\max\limits_i(\bar{X}_i^2)\geq t\right).$ When variables were gaussian, I used Chernoff inequality and obtain $pC_1e^{-C_2nt},$ where $C_1$ is pure constant and $C_2$ depend on variance.
I think that the first condition is related to sub-gaussian variables, but I don't know if it is equivalent, weaker, or stronger. Moreover, I don't know what the relationship is between the first and second conditions, if there is any meaningful one. If first condition implies sub-gaussian I can try probably use g.eneral Hoeffding’s inequality, but in the second case I completely have no idea.
