I am facing something that can be explained as a balls & urns problem. Suppose you have $B$ black and $W$ white balls inside an urn. They are randomly chosen, one by one, without replacement, and in each step $i$ we look at the proportion $\frac{B_i}{W_i}$ of balls already out of the urn.
In fact, it is not exactly the ratio, but $\frac{K + B_i} {H + W_i}$, where $K$ and $H$ are positive and known.
I need to be able to reasonably approximate tail values of the maximum and minimum distributions. That is, the maximum/minimum ratios that were reached in any step. The starting and ending values are known, what I need is to estimate the maximum deviations that can be reached with given low probabilities.
I did not know how to look for information regarding these distributions, this seems a simple problem and I guess it has been studied. If we are not looking at the maximum along time, but at the ratio in a specific step, then of course it is not hard: using combinatorics it is easy to calculate the exact (discrete) distribution.
Has someone heard about this?