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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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What you want to prove is obviously wrong and you misread the notes. The result is that if $X$ is sub-gaussian then $X^2$ is sub-exponential. –  Did Oct 7 '12 at 12:24
    
Maybe I was not clear. I agree with your statement. When $X'$ is sub-Gaussian, $X'^2$ is sub-exponential. But in my question $X$ is sub-exponential and I want tail bounds for $X^2$ in this case. –  krizna Oct 7 '12 at 18:27
    
Then there is no chance this can happen, as the simplest example shows. See my answer. –  Did Oct 7 '12 at 19:42
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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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Again, I never claimed that they are sub-exponential. Please take a look at the question again. Also for the interested(thanks to Prof. Vershynin's pointers): such a class of random variables are in the class $\psi_1$. One could handle them through something along the lines of Theorem 6.21 in probability in Banach spaces book. –  krizna Oct 10 '12 at 15:58
    
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.) –  Did Oct 11 '12 at 20:59
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