Suppose that $X,Y$ are two independent subGaussian RVs. Let $Z=XY$. Is $Z$ also subGaussian? Can someone provide any reference presenting some basic properties of subGaussian RVs. Thank you in advance!
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http://persomath.univmlv.fr/users/banach/SpringSummerSchool2011/VershyninRMTcourseIHP.pdf, http://cnx.org/content/m37185/latest/ and Folger and Rahut's Compressive sensing book all list some basic properties of sub Gaussian RV's. – Batman May 21 '14 at 13:49
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The product of two subgaussian random variables could be as subgaussian as not subgaussian.
To see that $Z = XY$ is not subgaussian in general when $X$ and $Y$ are you just need to take $X, Y \sim \mathcal{N}(0, 1/2)$. Write $$ XY = \frac 1 4 (X + Y)^2  \frac 1 4 (X  Y)^2 $$ Both $(X+Y)^2$ and $(XY)^2$ are distributed according to $\chi^2_1$(see here).
In fact, $\chi^2_p$ distribution is subexponential (not subgaussian) and linear combination of subexponential random variables is also subexponential.
As for $Z = XY$ being subgaussian take $X \sim \mathcal{N}(0, 1)$ and $Y \sim \mathcal{R}(1/2)$, where
$$ \mathcal{R}(p) = \begin{cases} +1, ~p \\ 1, ~1  p \end{cases} $$
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