Continuous Lower Probability-Based Models for Stationary Processes with Bounded and Divergent Time Averages

Abstract

We have undertaken to develop a new type of stochastic model for nondeterministic empirical processes that exhibit paradoxical characteristics of stationarity, bounded variables, and unstable time averages. By the well-known ergodic theorems of probability theory there is no measure that can model such processes. Hence we are motivated to broaden the scope for mathematical stochastic models. The emerging theory of upper and lower probability, a simple generalization of the theory of finitely additive probability, seems to provide a locus for this new modelling methodology. We focus our attention on the problem of the existence and construction of a lower probability $\underline{P}$ on the power set $2^X$ of a countably infinite product $X$ of a finite set of reals $X_0$, that is shift invariant, monotonely continuous along some class $\mathbf{M}$ of sets that includes the cylinder sets $\mathbf{C}$ and such that $\underline{P}(D^\ast) > 0$ where $D^\ast = \{\mathbf{x} = (x_i)_{i\in Z} \in X: (1/n)\sum^{n-1}_{i=0} x_i$ diverges as $(n \rightarrow \infty)\}$. We show that these constraints are incompatible when $\mathbf{M} = 2^X$, but when $\mathbf{M = C}$ we are able to construct such a lower probability. Most of our results extend to the case of a compact marginal space $X_0$.