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The original context of this problem is for a derivation of the lookdown model https://projecteuclid.org/journals/annals-of-probability/volume-24/issue-2/A-countable-representation-of-the-Fleming-Viot-measure-valued-diffusion/10.1214/aop/1039639359.full

The proof hangs on a very simple lemma, very easy to prove. I am looking for a name or reference for the lemma. It is simple yet surprisingly powerful.

Say you have a deck of cards, from which you draw a sequence of cards uniformly with replacement until an arbitrary condition is met (say, until the number of the card matches my Geiger counter.)

This is equivalent to the following process involving a permutation of the deck:

First, generate a random permutation of the cards (i.e., shuffle the deck).

Then pick cards from the top of the deck and keep a stack of previously drawn cards. suppose that by the time you completed the n th draw, the "previously picked cards" deck has c(n) cards. We then pick randomly from the "previously picked" pile with probability c(n)/52, and from the top of the original deck with probability 1-c(n)/52

Again, the proof is very easy, but the lemma has powerful and deep applications (see e.g. if you can understand the linked paper), so I was wondering whether it has a name.

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