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!


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