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Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas.

Consider the corresponding log-normal random variables: $Z_1 = \exp(X_1)$, $Z_2 = \exp(X_2)$.

Question: what is the distribution of the product of the two random variables, i.e., the distribution of $Z_1Z_2$?

If the normal random variables $X_1, X_2$ are independent, the answer is simple: we have $Z_1Z_2 = \exp(X_1+X_2)$ with the sum $X_1+X_2$ normal, hence the product $Z_1Z_2$ is still lognormal.

But suppose that $X_1, X_2$ are generally $not$ independent, say with correlation $\rho$. What can we say about the distribution of $Z_1Z_2$?

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Well, correlated or not, the distribution of X1+X2 is going to be normal, so I don't see any big difference. Only the variance of this is going to change according to the correlation.

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  • $\begingroup$ This is wrong. If the normal $X_1, X_2$ are not independent, or if does not exist a joint distribution, the sum $X_1+X_2$ is not normal. Take e.g. $X_2=-X_1$, it is normal but $X_1+X_2 = 0$. $\endgroup$ – RandomGuy May 14 '16 at 10:33
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    $\begingroup$ So that's just a limiting case, where the mean is zero and the variance is zero. That's still a normal distribution, albeit not a very interesting one. $\endgroup$ – Johnny Logan May 14 '16 at 18:07
  • $\begingroup$ I think you don't understand what normal distribution means. $\endgroup$ – RandomGuy May 14 '16 at 18:41
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    $\begingroup$ Why are you being hostile? The sum of two normal random variables is definitely a normal random variable. For instance, see here how they explicitly construct correlated variables from sums of uncorrelated variables. I can now add the two x's, and the sum is in their notation $x_1+x_2=(1+\rho)z_1+\sqrt{1-\rho^2}z_2$. This is a weighted sum of _un_correlated random variables. It is a normal random variable and in particular it has variance $2(1+\rho)$ - which includes the case you mention for $\rho=-1$. $\endgroup$ – Johnny Logan May 14 '16 at 21:42
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Even though this is an old post, I want to point out that one needs to be careful by saying "If $X_1$ is normal and $X_2$ is normal, then $X_1+X_2$ is normal, whatever the correlation $\rho$ between $X_1$ and $X_2$"

This statement is definitely wrong! The sum of two normals is normal if the dependency structure is normal (mathematically: if the copula is gaussian). However, if the dependence structure is not gaussian but has heavy tails (e.g. a Student-t copula) between $X_1$ and $X_2$, then $X_1+X_2$ will definitely not be normal distributed.

To come back to the question raised, the product of two lognormals will be lognormal if $(X_1,X_2)$ is a bivariate normal. Otherwise, this will not be the case!

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