Can any one give me an example of random vector $X = (X_1,\ldots,X_n)$ such that $$||X||_{\psi_2} \gg \max_i||X_i||_{\psi_2}.$$ Here $||X||_{\psi_2} = \sup_{x\in S^{n-1}}||\langle X,x\rangle||_{\psi_2}$ and $||X_i||_{\psi_2} = \inf\left\{t>0:E\left(\exp\left(X_i^2/t^2\right)\leq2\right)\right\}$. This is a question from high-dimensional probability and what I know is that if the coordinates of $X$ are independent then $||X||_{\psi_2} \leq C \max_i||X_i||_{\psi_2}$ for some constant $C$.
Thanks in advance!
Let $X_i:=Y$ for all $i\in\left\{1,\dots,n\right\}$. Then \begin{align} \left\lVert X\right\rVert_\psi&=\sup_{x\in S^{n-1}} \left\lVert \left\langle x,X\right\rangle\right\rVert_\psi\\ &=\sup_{x\in S^{n-1}} \left\lVert \sum_{i=1}^n x_iY \right\rVert_\psi\\ &= \sup_{x\in S^{n-1}}\left\lvert \sum_{i=1}^n x_i\right\rvert \left\lVert Y \right\rVert_\psi \end{align} and since $\sup_{x\in S^{n-1}}\left\lvert \sum_{i=1}^n x_i\right\rvert $ is achieved for $x_i=1/\sqrt n$, we get $$ \left\lVert X\right\rVert_\psi\geqslant \sqrt n\max_{1\leqslant i\leqslant n}\left\lVert X_i \right\rVert_\psi. $$