Ergodic theorist (occasional) here. Ergodic theory is basically studying situations in which the time average of (almost every) trajectory equals the space aver

Ergodic theorist (occasional) here. Ergodic theory is basically studying situations in which the time average of (almost every) trajectory equals the space average of the system. This is a widely studied area in dynamical systems. I am squeamish about such popular accounts. An old account which is readable even today, is by Birkhoff [1]. About Kelly-Bernoulli criterion, the readable account is the book [2]. [1] https://doi.org/10.1080/00029890.1942.11991212 https://doi.org/10.1080/00029890.1942.11991212 [2] "Fortune's Formula" by William Poundstone. https://archive.org/details/fortunesformulau00poun https://archive.org/details/fortunesformulau00poun