= Repeat-until-success coding over an erasure channel
{title2=$M_j\sim\operatorname{Geom}(1-q),\quad n/(M_1+\cdots+M_n)\to1-q$}
With <channel feedback>, resend each message <bit> through a <binary erasure channel> until a non-erased copy arrives, then move to the next <bit>. The <channel decoder> deletes erasure symbols, giving the message without error whenever transmission completes. For $q<1$, the waiting times have an <independent and identically distributed> <geometric distribution> with <expectation> $1/(1-q)$, so every finite message completes <almost surely>. The <strong law of large numbers> gives forward <code rate> $n/\sum_{j=1}^nM_j\to1-q$, matching <channel capacity>. At $q=1$, transmission never completes. Independent erasure indicators, rather than independent message <bits>, are the essential hypothesis.
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