the product of Gaussian

  • with any hypothesis, the product of Gaussian is not Gaussian, take Y=X as an example
  • if X, Y are independent, it is still not Gaussian. Consider
Z = \frac{X^2-Y^2}{2} = \frac{X-Y}{\sqrt 2} \frac{X+Y}{\sqrt 2}

then Z is the product of two independent Gaussian, but the characteristic function of Z is

\begin{align*} \varphi_Z(t) &= E\exp\left(\frac{X^2-Y^2}{2}\right) \\ &= E\exp\left(1-2i\frac t2\right)^{-1/2}\exp\left(1-2i\frac{-t}{2}\right)^{-1/2} \\ &= (1-it)^{-1/2}(1+it)^{-1/2}=(1+t^2)^{-1/2}\,. \end{align*}

refer to Is the product of two Gaussian random variables also a Gaussian?

functions of independent random variables

it is trivial with sigma algebra.

refer to Functions of Independent Random Variables