Distribution of the product of two LogNormals [duplicate]

Question

Reading the wiki I saw the statemeent that: "The distribution of the product of two random variables which have lognormal distributions is again lognormal. This is itself a special case of a more general set of results where the logarithm of the product can be written as the sum of the logarithms"

This enters in contadiction with my previous undernstanding( mainly the need of uspposing a bivariate lognormal distribution in order to obtain such a result)

Does anybody have some insight on the rational for such an assumption on the wiki?

Thank you a lot in advance,

Answer 1

If $X \sim YZ$, then $\log X \sim \log Y + \log Z$. If Y and Z are both log-normal, the two random variables $\log Y$ and $\log Z$ are both normal, hence $\log X$ is normal.

Edit: as pointed out by @whuber, $\log Y$ and $\log Z$ must also be jointly normal for their sum to be normal; one simple reason that they might be jointly normal is if they are independent, which implies joint normality. Joint normality of two distributions is not equivalent to those distributions being independent and separately normal -- a simple exception would be the dependent variables $W$ and $Q=W$, whose sum is just $2W$ and is also normal. However, the definition of being jointly normal is simply that that $a \log Y + b \log Z$ is normal for all $a,b \in \mathbb{R}$, and so proving joint normality in full generality is no easier than proving the necessary special case where $a=1$ and $b=1$.