In this paper Ole Peters and Bill Klein clarify two important issues in geometric Brownian motion. They show analytically, using extreme-value theory, that the time-average growth rate for a finite ensemble is the same as the time-average growth rate for a single system. Next, using techniques from continuous ordering and the theory of hysteric phase transitions, they investigate the trade-off between finite observation times and finite ensemble sizes. Ensemble-average behaviour is found to dominate for a time that grows logarithmically slowly with the ensemble size. In other words, even in large ensembles, ensemble averages become irrelevant after a short time. This analysis extends Sid Redner’s work on the multiplicative binomial process to geometric Brownian motion and adds a correction term.
The intuitive simple arguments in Sid’s paper cannot be immediately applied to geometric Brownian motion because here, even after a short time, the largest possible value of the process is infinity.