# Distribution of a normalized multivariate log-gaussian process

Let $$p, q \in \mathbb{N}$$ and $$Z$$ a gaussian process over $$D \subset \mathbb{R}^{p+q}$$ : $$Z(\cdot) \sim \mathcal{GP}(\mu(\cdot), k(\cdot,\cdot))$$.

For $$(x_1, \ldots, x_p, y_1, \ldots, y_q)^T \in D$$, we define an associated process $$W$$ as

\begin{align} & W(x_1, \ldots, x_p, y_1, \ldots, y_q) \\[10pt] = {} & \frac{\exp[Z(x_1, \ldots , x_p, y_1, \ldots, y_q)]}{\int_{u_1}\cdots\int_{u_q} \exp[Z(x_1, \ldots, x_p, u_1, \ldots, u_q)] \,du_1 \cdots \,du_q}. \end{align}

How to find the distribution of $$W$$ ?

I considered the following lead : if $$Z$$ has continuous paths, then we can see the integral as the limit of a Riemann sum. However, I found that there was no closed-form formula for the distribution of a sum of dependant log-normal variables...

I would be grateful if some of you could give me some hints, other leads I can follow or for any related answer :)