Let $\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$ and $\mathbf{Y} = \text{exp}(\mathbf{X})$. If $Y_i$ is one of the components of $\mathbf{Y}$, what is the distribution of $\frac{\mathbf{Y}}{Y_i}$?

The answer to this question explains how to find the distribution of the ratio of log-normal random variables, however I am struggling to derive a result for the above multivariate case.


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I think I solved my own problem. Rewriting $\log \frac{\mathbf{Y}}{Y_{i^*}}$ as

\begin{align*} \log \frac{\mathbf{Y}}{Y_{i^*}} &= \log \mathbf{Y} - \log Y_{i^*}\mathbf{1} = \mathbf{X} - X_{i^*}\mathbf{1} = (I - B)\mathbf{X}\\ \text{where } B_{i,j} &=\begin{cases} 0 & j \neq i^*\\ 1 & j = i^*\end{cases}, \end{align*} we see that we simply need to find the distribution of an affine transformation of a multivariate normal. Using the known result for this, we obtain

\begin{equation*} \log \frac{\mathbf{Y}}{Y_{i^*}} \sim \mathcal{N}((I - B)\boldsymbol{\mu}, (I-B)\Sigma(I-B)^T) \end{equation*}


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