# Correlation between the reciprocal of lognormal random variables

Let $X_1$ and $X_2$ be identically distributed correlated lognormal random variables:

$$X_1,X_2 \sim \ln \mathcal{N}(\mu_X,\sigma_X^2)$$

(such that their logs are bivariate normal).

The correlation between $X_1$ and $X_2$ is given by:

$$\text{corr}(X_1,X_2) = \rho_X$$

Let $Y_1$ and $Y_2$ be the reciprical of these random variables:

$$Y_1 = \frac{1}{X_1}, Y_2 = \frac{1}{X_2}$$

Then $Y_1$ and $Y_2$ will also be lognormally distributed according to:

$$Y_1,Y_2 \sim \ln \mathcal{N}(-\mu_X,\sigma_X^2)$$

The correlation between them is given by $\rho_Y$:

$$\text{corr}(Y_1,Y_2) = \rho_Y$$

Can an expression for $\rho_Y$ be written in terms of $\rho_X$?

• it's easy in the case where $(\log X_1,\log X_2)$ are bivariate normal. Did you intend that situation or did you mean for it to be a general question? – Glen_b Aug 2 '16 at 13:23
• I think that I intend this to be the case. I am thinking about the reciprocal of a lognormal random process (which is obtained through the exponentiation of a multivariate Gaussian). I know what the marginal distribution of the reciprocal lognormal will be but I am not sure about the correlation structure. I chose lognormal since the reciprocal transformation is nice. The reciprocal of a lognormal variable is lognormal. I believe other distributions are not easy to work with when the reciprocal is taken. – 7Jack Aug 2 '16 at 14:18

If you have two variables, $Z_1$ and $Z_2$ that are bivariate normal, then the correlation of two corresponding lognormals is a simple function of the correlation of the corresponding normals and their standard deviations. For example, mpiktas' answer to this question gives the result.
Now $-Z_1$ and $-Z_2$ are bivariate normal with the same variances and correlation as $Z_1$ and $Z_2$, so $Y_1=e^{-Z_1}$ and $Y_2=e^{-Z_2}$ must have the same correlation as $X_1=e^{Z_1}$ and $X_2=e^{Z_2}$ do.