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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.

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    The$n$ there is $n$o ch$a$n$c$e this can happen, as the simplest example shows. See my answer.2012-10-07

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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.

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    You asked for an exponential control of the tail of X^2 when X follows a subexponential distribution, in the question itself (quote: *But I need exponential tail bounds for X^2*) and in a comment, and I showed that such a control cannot exist in general. If this is not your question, you might wish to rephrase. (Note that the juvenile admonestation *Please take a look at the question again*, when irrelevant, could alienate you potential answerers.)2012-10-11