Have there been earlier descriptions of the following compound distribution?
Compounding a Gaussian distribution with variance distributed according to the absolute value or square of another Gaussian distribution:
$$ f(y) = \int \phi(x)\frac{\phi(y/\vert x\vert)}{\vert x \vert} dx$$
This distribution would look like this:
and it possibly relates to a solution for this question Limiting distribution of $\frac1n \sum_{k=1}^{n}|S_{k-1}|(X_k^2 - 1)$ where $X_k$ are i.i.d standard normal
compGaus <- function(x) {
dy <- 0.1
y <- seq(-16,16,dy)
out <- 0
for (ys in y) {
var = ys^2
dens = dnorm(ys,0,1)
if (var>0) {
out = out + dens*dnorm(x/var^0.5,0,1)/var^0.5*dy
}
}
out
}
compGaus <- Vectorize(compGaus)
x <- seq(-2,2,0.1)
plot(x,compGaus(x),log="")