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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?

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    $\begingroup$ Try an example with $\xi_i$ being fair coin flips and deleting the first heads. It should be easy to check that $\xi_{i_1}$, $\xi_{i_2}$ are neither independent nor identically distributed. One simple condition under which the subsequence is iid would be if the subsequence is selected independently to the original sequence. $\endgroup$ – Nate Eldredge Nov 16 '16 at 0:03
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    $\begingroup$ For the rule "remove the first heads if there is one, otherwise remove any tail", the possibilities hhh hht hth htt thh tht tth ttt lead to hh ht th tt th tt tt tt. So the first element of the subsequence has a 3/4 chance of tails and the second element has only a 5/8 chance of tails: not identically distributed, and not independent either. $\endgroup$ – Matt F. Nov 16 '16 at 2:21

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