I have a question regarding coupling a random variable with the same random variable conditioned on a particular event. The example I will be discussing is taken from "Coupling and Poisson Approximation" by Janson.
Let $r$ balls be thrown independently of each other into $n$ boxes labelled $1,...,n$. Let $p_i$ denote the probability of hitting box $i$. Defined the indicator $I_i = I(\text{box $i$ is empty})$, and let $W=\sum_{i=1}^n I_i$. For $j\neq i$, we may create a coupling of $I_i$ and $I_i | I_j=1$ by removing all balls from box $j$ and redistributing them according to the original distribution, repeating until box $j$ is empty.
While seems obvious that this should be a coupling of $I_i$ and $I_i | I_j=1$, it is not clear to me how I would write down the joint distribution for the coupling if I wanted to rigorously verify that, especially since in the conditional space, our probabilities are normalised by $\mathbb{P}(I_j=1)$. There are several examples of this type in the paper, so I'm hoping if I can understand this simpler one, I can understand the others. Thanks.
It's equivalent but easier to see if you consider "fresh balls" from a countable sequence instead of reusing the old ones that were thrown in box $j$. Here is a coupling. Consider $k$ boxes and a given box $j\le k$ of interest. Consider a countable ordered sequence of iid balls $B_1,B_2, B_3,...$ thrown into the boxes.