I am looking for a weak convergence of marginals result, in the following type of situation. References to 'related' situations are also very much appreciated.

I have a collection of affine hyperplanes $H_{1}, H_{2}, \ldots$ and variables $X_{1}, X_{2}, \ldots$ such that each variable appears in at most, say, 10 hyperplanes and each hyperplane has at most 10 variables. In my situation, all of the coefficients for the variables were negative while the offset was always positive, which made things easier.

Now, look at the finite-dimensional convex body $K_{n}$ in $\mathbb{R}^{n}$ containing the origin, bounded by $1 \geq X_{i} \geq 0$ and all of the hyperplanes that only contain variables $X_{i}$ with $i \leq n$. If I draw uniformly at random from $K_{n}$, I get some marginal distribution on $X_{1}$.

My question is, under what conditions does this marginal distribution have a limit, even when the limit of volume($K_{n}$) is 0?

2 EDITS: As pointed out by Ricky Demer in the comment below, we sometimes have convergence to deterministic limiting distributions by forcing some of the values to e.g. eventually be 0. I think I'm mostly interested in cases where the limiting marginals are all non-deterministic..