# How to prove that the centering inequality for the sub-gaussian norm does not hold

Specifically, define the sub-gaussian norm for a r.v. X as

$$\|X\|_{\psi_2}=\inf\{t>0:e^{X^2/t^2}\leq 2\}.$$

How do we prove that the centering inequality with $C=1$ does not hold in general? i.e. $$\|X-\mathbb{E}X\|_{\psi_2}\leq C\|X\|_{\psi_2}.$$ A counter example would work too. Thanks in advance!