A series of matches are held between n identical competitors. Each is won by one of the n with equal probability (no ties). I'm looking for a probabilistic description of the outcome when looking at the first player to win 1, 2, ... matches.
For example, if player #3 is the first player to win 400 matches, she has a better than 1/n chance to be the first player to win 401 matches.
In particular: how many matches need to be played before some player wins k? (The dominant term is of course kn but what is the next (negative) term?) How often does the lead change places? (I expect infinitely often, but with decreasing frequency... maybe sqrt-ly many times?)
I have a decent math background but have not studied any probability since a basic undergrad class.
Related question: How long until everyone is in the lead?