For a project, I am looking into central limit theorems for stationary mixing random fields. I found many results for the asymptotic distribution of partial sums, such as that of Bolthausen (1982). However, in my application, I am looking for a "conditional" version of these CLTs.
For example, consider a real valued stationary mixing random field $X_\rho, \rho \in \mathbb{Z}^d$ over the regular $d$ dimensional lattice, consisting of two random variables: i.e. $X_\rho = (V_\rho, W_\rho)$. Let $\Lambda_n$ be a fixed sequence of finite subsets of $\mathbb{Z}^d$ that increase to $\mathbb{Z}^d$ and define $S_n = \sum_{\rho \in \Lambda_n} V_\rho$. I am interesting in the asymptotic conditional distribution $ \lim_{n\rightarrow \infty}\Pr[S_n <= s \mid W_\rho = w]$. Standard results only seem to deliver the unconditional distribution $ \lim_{n\rightarrow \infty}\Pr[S_n <= s]$.
Do you have any suggestions on how to proceed? References the the literature are also greatly appreciated.