Say I have a bunch of IDD random variables $\mathbb X_n$ with some distribution strictly in $\mathbb R^+$ (realistically we could even limit it to be on a compact subset, say $[a,b] \subset \mathbb R^+$), furthermore let us say we from this build
$$\mathbb S_n = \sum_{k=0}^n \mathbb X_k$$ and then build something like $$p(\mathbb X_k) = \frac{1}{\mathbb X_{k+1}}\int_{\mathbb S_{k}}^{\mathbb S_{k+1}}f(\tau)d\tau$$
Something like a "random sampling" procedure, $\mathbb X_k$ could be interpreted as some kind of exposure.
But to the question... How could we analyze something like this? I guess we will need to take help at least from both probability and analysis? Also any references to work which deals with this would be welcome.