Probability of pr. – The norm of the isotropic subagusian vector may not be subgalusian.

Suppose $ X $ it is an isotropic sub-Gaussian $ n $three-dimensional vector (ie $ EXX ^ T = I_n $, and for any unit vector. $ u $,$ | left | _ { psi_2} le K $). People say that $ | X | _2- sqrt n $ It may not be sub-Gaussian. But I have not found an opposite example.

When $ X $ is a uniform distribution of the ball or a uniform distribution of the hypercube, you can prove that $ | X | _2- sqrt n $ It is sub-Gaussian. On the other hand, if $ X_i $ They are independent, the proposition is also true.

Can someone show a counter example? Thank you!