If they would allow me to play a +EV game I can crush them even if they had infinite bankroll by varying my betsize. The 'random walk' argument only works for fixed bet size.
Turns out, you have to go surprisingly high for this to be true/effective on the long run. And the casino will cut you off way below the limit (i.e. there's of course a maximum bet). Plus, with this strategy, each win is a small one... but when you lose (hitting the max bet), the loses are significant.
This is the St. Petersburg paradox: https://en.wikipedia.org/wiki/St._Petersburg_paradox
If a business offered a positive EV gambling game, it wouldn't be called a casino and would only last as long as the venture capital lasted.
http://www.slotsmamma.com/blog/how-do-you-find-full-pay-deuc...
There are a few catches though. Mostly if you play perfectly (which is not simple) the most you can expect to make is less than eight dollars an hour.