Let $x$ be a continuous random variable with zero mean and zero skew. What are the conditions under which we can say that $x$ can be expressed as the product $z y$ where $z$ is a standard normal and $y$ is a strictly positive random variable independent of $z$? I don't think, for example, that unimodality is required, but I think we do require that the CDF of $x$ be continuous, say (no point masses). Are there other conditions?
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$\begingroup$ OK, duh, well $x$ has to have zero odd centered moments, which is a bit of a restriction. $\endgroup$– Steven PavCommented Dec 28, 2016 at 21:38
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$\begingroup$ You should realy add that info to the OP not as a comment! $\endgroup$– kjetil b halvorsenCommented Dec 28, 2016 at 21:42
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1$\begingroup$ The moments don't necessarily exist, but the distribution is symmetric about $0$. $\endgroup$– Robert IsraelCommented Dec 28, 2016 at 22:08
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1 Answer
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The characteristic function of $x$ is $\varphi(s) = \mathbb E[e^{isx}] = \mathbb E[e^{-s^2 y^2/2}]$. The fact that $\varphi(s) \to 0$ as $s \to \pm \infty$ tells you that $x$ is continuous.
answered Dec 28, 2016 at 22:09