# Log-normal ratio implies numerator/denominator are log-normal

Let $$X$$ and $$Y$$ be two positive random variables defined over $$(\Omega,\mathscr{F},\mathbb{P})$$. We know that if they are both log-normal then the random variable $$Z$$ defined as: $$Z:=\ln\frac{X}{Y}$$ is normally distributed.

Is the converse true? Namely if $$Z$$ is a normal variable defined as above, can we conclude $$X,Y$$ are both log-normal? Equivalently, if the ratio $$X/Y$$ is log-normal, does it imply $$X,Y$$ are so too?

• Definitely not, because you make no assumptions about how $X$ and $Y$ are jointly distributed. So just take $Z$ and independently of it choose $Y$ to have any positive distribution you like and define $X=Ye^Z.$ If you intend that $X$ and $Y$ be independent, then please edit your post to indicate that.
– whuber
Jul 18, 2023 at 13:36
• You may as well remove the lognormal part, by considering $V=\log(X), W=\log(Y)$ and so ask about $Z=V-W$ and normal distributions Jul 18, 2023 at 13:53

Yes. If $$X$$ and $$Y$$ are independent positive variables and $$X/Y$$ is lognormal, then $$\ln(X/Y)=\ln X +(- \ln Y)$$ is normal. So by Cramer’s decomposition theorem both $$\ln X$$ and $$-\ln Y$$ are normal, and $$X$$ and $$Y$$ must be lognormal.