Recently, I fell in love with the pointwise/elementwise/componentwise/Hadamard/Schur functions and compositions of random vectors such as Hadamard squares and products of random vectors. Here is one example:

Normal approximation to the pointwise/Hadamard/Schur product of two multivariate Gaussian/normal random variables

Unfortunately, that does not look like a very popular topic. I even read on MO that « nobody cares about the Hadamard products of (random) vectors”. I do because it appears that they are of primary interest in some practical applications.

Hence, I’m looking for nice, introductory references about the general theory of Hadamard functions and compositions of random vectors. Special topics of interest include:

  • Multivariate analogue/generalization of the normal product distribution for the Hadamard product $X \circ Y$ of central or non-central Gaussian random vectors;
  • Multivariate analogue/generalization of the central and non-central chi-squared distributions for the Hadamard square ${X^{ \circ 2}} = X \circ X$ of a central or non-central Gaussian random vector;
  • Multivariate distribution of the Hadamard exponential of a Gaussian random vector;
  • ...

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