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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?

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    $\begingroup$ 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. $\endgroup$
    – whuber ♦
    Commented Jul 18, 2023 at 13:36
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    $\begingroup$ 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 $\endgroup$
    – Henry
    Commented Jul 18, 2023 at 13:53

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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.

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