Let $X \sim \mathcal{N}\left( {{\mu _x},\sigma _x^2} \right)$ and $Y \sim \mathcal{N}\left( {{\mu _y},\sigma _y^2} \right)$ be two univariate and independent Gaussian/normal random variables and let $Z = XY$ be their product.

It is well-known that the probability distribution of $Z$ can be approximated by the normal distribution $\mathcal{N}\left( {{\mu _x}{\mu _y},\sigma _x^2\sigma _y^2 + \mu _x^2\sigma _y^2 + \mu _y^2\sigma _x^2} \right)$ when $\frac{{{\mu _x}}}{{{\sigma _x}}}$ and $\frac{{{\mu _y}}}{{{\sigma _y}}}$ are large. See for instance An approach to distribution of the product of two random variables.

I’m looking for similar results in the multivariate case : $X \sim {\mathcal{N}_d}\left( {{\mu _x},{\Sigma _x}} \right)$ , $Y \sim {\mathcal{N}_d}\left( {{\mu _y},{\Sigma _y}} \right)$ and $Z$ is the pointwise/Hadamard/Schur product of $X$ and $Y$ , i.e. $Z = \left( {{x_1}{y_1},{x_2}{y_2},...,{x_d}{y_d}} \right)$.

More generally, I’m looking for references dealing with the properties of the pointwise/Hadamard/Schur products of multivariate random variables (as a generalization of the scalar product of univariate r.v.s).


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