It is clear the product of normal distributed variables is not normal distributed. For example, if $X \sim N( \mu_1,\sigma_1^2)$, $Y \sim N( \mu_2,\sigma_2^2)$, then $XY$ does not has the distribution of $ N( \mu_1 \mu_2,\mu_1^2 \sigma_1^2+\mu_2^2\sigma_1^2)$.
I have been told that even if the distribution of $XY$ is not normal distribution, the distribution of $XY$ is near to normal distribution, when $\mu_1$ and $\mu_2$ are not so small, $\sigma_1$ and $\sigma_2$ are not so big. Is it true?
Try following R code:
n1<-rnorm(10000,0,.005) n2<-rnorm(10000,0,.005) n<-n1*n2 d<-density(n) plot(d,lwd=2) x<-par('usr') dn<-dnorm(d$x,mean=mean(n),sd=sd(n)) x<-seq(x,x,length.out=length(dn)) lines(x,dn,col=2,lwd=2) legend('topright',legend=c('Estimated density','Normal distribution'),lwd=2,lty=c(1,1),col=c(1,2))
It seems only when two conditions are both meet, the distribution is near normal. Is there any theoretical analysis?