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Keywords:
mormal approximation; Gaussian scale mixture; Plancherel theorem
mormal approximation; Gaussian scale mixture; Plancherel theorem
Summary:
For a given positive random variable $V>0$ and a given $Z\sim N(0,1)$ independent of $V$, we compute the scalar $t_0$ such that the distance in the $L^2(\mathbb{R})$ sense between $Z V^{1/2}$ and $Z\sqrt{t_0}$ is minimal. We also consider the same problem in several dimensions when $V$ is a random positive definite matrix.
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