Let $X$ be a sub-exponential random variable as defined in section 5.2.4 of Roman Vershynin's notes available here: http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf . In that case, there exists exponential tail bounds for $X-\mathbb{E}X$. But I need exponential tail bounds for $X^2-\mathbb{E}X^2$. Any ideas or pointers to relevant literature will be appreciated.
[Math] Tail bounds for square of sub-exponential random variable
probability theory
Best Answer
Loosely speaking, $X$ is subexponential if $\mathbb P(X\geqslant x)\leqslant\mathrm e^{-cx}$ for some positive $c$, for every $x$ large enough. Then $Y=X^2$ is such that $\mathbb P(Y\geqslant x)=\mathbb P(X\geqslant \sqrt{x})\leqslant\mathrm e^{-c\sqrt{x}}$ for every $x$ large enough. Hence there is no reason for $Y$ to be subexponential.
The simplest example might be when $X$ is standard exponential, then $\mathbb P(X\geqslant x)=\mathrm e^{-x}$ for every nonnegative $x$, hence $X$ is subexponential, and $\mathbb P(Y\geqslant x)=\mathrm e^{-\sqrt{x}}$ for every nonnegative $x$, hence $Y$ is not subexponential.