# 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!