For a random variable $X$ and positive integers $m,k$ the tail bound is: $$P(|X|>t) \le \exp(- m t^{2/m} )$$ And for any $k\in\mathbb{N}^{+}$ the moments are given by: $$E X^{2k+1} = 0$$ $$E X^{2k} \le (k)^{m k}$$ If we center $X^2$ by $Y:=X^2 - E(X^2)$ , what can be said about the lower its tails? $P(|Y|\ge t) = P(|X^2-E(X^2)|\ge t)$?
For the special case that $m=1$ and $m=2$, the tail bound implies that $X$ is sub-Gaussian and sub-exponential respectively, and one can use its machinery to derive some bounds. But I'm not sure what is the best way to derive the tail bounds for this case.
More context: this is a followup question on this MO question. As you can see, the answer derives the tail bound and moment bound mentioned here (discarding some constants). I thought there must be enough information in the moments to prove something about the tails of centered variable, which is why I didn't give the full context (I could be wrong thought?!)