Waiting time for a word in independent uniform symbols

Let a fixed word $w$ be sampled from an alphabet of $q$ symbols by independent uniform draws, and let ${\tau }_w$ count draws through its first occurrence. If $B(w)$ is the set of lengths of the borders of a word, including the full length, thenShorter borders contribute because a failed or completed match can retain a suffix that is already a prefix of the next match.