Let $\xi_1,\ldots,\xi_n$ be an iid sequence of random variables. If we take a sub-sequence $\xi_{i_1},\ldots,\xi_{i_k}$ with *constant* indices $1\leq i_1 <\ldots <i_k\leq n$, then the sub-sequence will clearly remain iid. If, however, the sub-sequence is selected randomly, in a data-dependent way, this may not be true.

A simple example: select the first two *different* entries, in the order they occur. (If all entries are equal, then just select the first two.) Then, generally, they will not form an iid sub-sequence, as they are forced to take different values, unless all entries are equal.

**Question:**

Consider now the following selection rule: Let $H$ be a subset of the common range. Remove the first hit to $H$ from the sequence, and keep the rest in their original order. (If possibly there is no hit to $H$, then the whole sequence is kept.) Will the obtained sequence remain iid?

More generally, is there any result that characterizes which random selection rules lead to an iid sub-sequence?