Suppose that $N\sim(0,1)$ and I have two functions $f$ and $g$.
Are there any theorems or techniques for comparing how close are distributions of $f(N)$ and $g(N)$?
For example, it is known that sum of lognormal random variables $ae^{bN}+ce^{dN}$ can be approximated by one lognormal variable $ue^{vN}$. Is it possible to find how good approximation is?
Thank you.