Before revision: Geometric Brownian motion is non-stationary. It is non-ergodic in the sense that the time-average growth rate observed in a single realization differs from the growth rate of the ensemble average. We prove that the time-average growth rate of averages over a finite number, N, of realizations is independent of N. A stability analysis shows that the time at which such averages begin to deviate from ensemble-average behavior increases logarithmically with N .
After revision: Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by non-ergodicity, which can lead to ensemble averages exhibiting exponential growth while time averages suffer collapse. A common tactic for dealing with this difference is diversification. In this letter we show that diversification will not solve the problem but only delay the collapse .