I am self-studying Roman Vershynin's High-Dimensional Probability and am trying to solve Exercise 3.4.10 of the book. It asks to show that the concentration inequality in Theorem 3.1.1 in the book may not hold for a general isotropic sub-gaussian random vector (a random vector $X$ is called isotropic if $EXX^t = I_n$).
To be specific, the Theorem 3.1.1 states that if $X = (X_1, X_2, \cdots, X_n)$ be a random vector with independent sub-gaussian coordinates $X_i$ that satisfies $EX_i^2 = 1$, then $$\big\| \|X\|_2 – \sqrt{n} \big\|_{\psi_2} \leq CK^2$$
where $K = \max_i \|X_i\|_{\psi_2}$ and C is an absolute constant, i.e. does neither depend on $X$ nor $n$.
The exercise intends to show that independence of the coordinates of $X$ is an essential assumption in the Theorem 3.1.1.
I have tried to find such an example by modifying some known isotropic random vectors such as the coordinate distribution and uniform distribution on the Euclidean ball. But, it seems like modified version of these distributions won't yield one. I appreciate any advice that might work.
Best Answer
Figured $X = \sqrt{2} \varepsilon \psi$, where $\varepsilon$ is Bernoulli (1/2) and $\psi$ ~ Uniform($\sqrt{n}S^{n-1}$) are independent, suffices.
This is taken from the solution to Exercise 6.3.6 provided here.
https://zhuanlan.zhihu.com/p/338822722