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Suppose that $X,Y$ are two independent sub-Gaussian RVs. Let $Z=XY$. Is $Z$ also sub-Gaussian? Can someone provide any reference presenting some basic properties of sub-Gaussian RVs. Thank you in advance!

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The product of two sub-gaussian random variables could be as sub-gaussian as not sub-gaussian.

To see that $Z = XY$ is not sub-gaussian 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 $(X-Y)^2$ are distributed according to $\chi^2_1$(see here).

In fact, $\chi^2_p$ distribution is sub-exponential (not sub-gaussian) and linear combination of sub-exponential random variables is also sub-exponential.

As for $Z = XY$ being sub-gaussian 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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