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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**.2012-10-07
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    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.2012-10-07
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    Then there is no chance this can happen, as the simplest example shows. See my answer.2012-10-07

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