# Find an example of a random vector $X$ with its sub-gaussian norm much larger than any of its coordinates'.

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 $$C$$.

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 $$\left\lVert X\right\rVert_\psi\geqslant \sqrt n\max_{1\leqslant i\leqslant n}\left\lVert X_i \right\rVert_\psi.$$