0
$\begingroup$

My question is similar to this one The product of two lognormal random variables, but I'd like to do it with three lognormals. The referenced question cannot be applied twice since the correlation between $X_1X_2$ and $X_3$ is not known.

Let $X_1$, $X_2$ and $X_3$ be three normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$, $X_2\sim N(\mu_2, \sigma^2_2)$ and $X_3\sim N(\mu_3, \sigma^3_2)$. The tri-variate normal distribution $(X,Y,Z)$ has a correlation structure with $\rho(X_1,X_2)=\rho_{X_1X_2}$, $\rho(X_1,X_3)=\rho_{X_1X_3}$ and $\rho(X_2,X_3)=\rho_{X_2X_3}$.

Consider the corresponding log-normal random variables: $Z_1 = \exp(X_1)$, $Z_2 = \exp(X_2)$ and $Z_3 = \exp(X_3)$.

Question: what is the distribution of the product of the three random variables, i.e., the distribution of $Z_1Z_2Z_3$?

It's easy to compute the distributions of $Z_1Z_2$ using the referenced question, but I'm stuck because I don't know the correlation between $X_1X_2$ and $X_3$.

$\endgroup$
  • $\begingroup$ Just apply the referenced answer twice. Note that it requires only the correlations among the $X_i,$ not among the $Z_i.$ $\endgroup$ – whuber Oct 31 '19 at 14:21
  • $\begingroup$ @whuber But how do you get the correlation between $X_1X_2$ and $X_3$? $\endgroup$ – John Doe Oct 31 '19 at 14:26
  • $\begingroup$ That's irrelevant, because $\log(Z_1Z_2Z_3)=X_1+X_2+X_3,$ not $X_1X_2+X_3.$ The whole point of the duplicate is that $X_1+X_2+X_3$ has a Normal distribution and so $Z_1Z_2Z_3$ has the corresponding lognormal distribution. $\endgroup$ – whuber Oct 31 '19 at 15:11