# Any sub-vector of an isotropic sub-gaussian random vector is an isotropic sub-gaussian random vector.

Any sub-vector of an isotropic sub-gaussian random vector is an isotropic sub-gaussian random vector.

Note that a random vector $$x\in\mathbb{R}^n$$ is sub-gaussian if for any $$a\in\mathbb{R}^n$$, $$a\neq0$$, we have $$x^Ta$$ is a sub-gaussian random variable. Additionally, a random vector $$x\in\mathbb{R}^n$$ is isotropic if $$Ex=0$$ and $$\operatorname{Cov}(X)=I_n$$. Equivalently, $$x$$ is isotropic if $$E[(x^Ta)]=\|\ a \|_2^2 \quad \quad \forall a\in\mathbb{R}^n.$$ Let $$x$$ be an isotropic sub-gaussian random vector and let $$y$$ be a sub-vector of $$x$$, so that if $$x=(x_1,\dots,x_n)$$ then $$y=(y_{J_1},\dots,y_{J_k})$$, where $$J=\left\{J_1,\dots,J_k\right\}\subset\left\{1,\dots,n\right\}$$.

I'm having trouble arriving to the desired result.