The product of two lognormal distributions may not be a lognormal If two lognormal random variables $X, \:Y$ are neither independent nor jointly normally distributed, the product $XY$ may not be lognormally distributed.
Where can I find the explanation for the 'may not be' part in the second statement?

Suppose $X\sim LN(\mu, \sigma^2)$  is log-normally distributed.
Then its reciprocal $\frac{1}{X}\sim LN(-\mu, \sigma^2)$  is also log-normally distributed.
But of course $X\cdot\frac{1}{X}=1$ is a degenerate point mass at $1$ and not log-normal any more.
The problem is that $X$ and $\frac{1}{X}$ are not independent.
