[AaronBrown]

I'm not sure if everything has been resolved or not.

The long run average winning percentage converges to the expected win rate.

The long run number of wins "diverges" in the sense that the expected absolute difference between the current number of wins and future number of wins increases, the farther you look into the future.

However, the long-run number of wins can also be said to neither converge nor diverge. It's not a question of mathematical definitions, it's a question of English. If you flip a fair coin and win $1 for heads and pay $1 for tails, your expected wealth after any number of future flips is equal to your current wealth. It does not tend to zero or anything else.