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Prove that any subvector of an isotropic Sub-Gaussian random vector is an isotropic Sub-Gaussian random vector.

Let $X \in \mathbb{R}^{n}$ be an isotropic Sub-Gaussian random vector. Then we know that $EX=0$, $Cov(X)=I_n$, and $E[(X^Ta)^2)]=||a||^{2}_{2}$ for all $a\in \mathbb{R}^{n}$. Also, for any $a\in \mathbb{R}^{n}, a \ne 0, X^Ta$ is a Sub-Gaussian random variable.

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  • $\begingroup$ And what have you tried? $\endgroup$ – Don Thousand Oct 15 '18 at 20:06

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