I have two normally distributed random variables (zero mean), and I am interested in the distribution of their product; a normal product distribution.

It's a strange distribution involving a delta function.

What is the variance of this distribution - and is it finite?

I know that

$Var(XY)=Var(X)Var(Y)+Var(X)E(Y)^2+Var(Y)E(X)^2$

However I'm running a few simulations and noticing that the sample average of variables following this distribution is not converging to normality - making me guess that its variance is not actually finite.