Finite-Dimensional Distributions and Tail Behavior in Stationary Interval-Valued Probability Models

Abstract

We consider the relationship between the finite-dimensional distributions of a stationary time series model and its asymptotic behavior in the framework of interval-valued probability (IVP), a simple generalization of additive probability measures. By Caratheodory's theorem, the specification of a countably additive probability measure on the algebra of cylinders $\mathscr{C}$ uniquely defines its behavior on $\sigma(\mathscr{C})$ (containing the tail events). If the measure is stationary, then the ergodic theorem indicates that its extension assigns zero probability to the tail event consisting of all sequences for which the time averages diverge (the divergence event). This link between the marginals and the tail behavior is no longer valid in IVP, and we can reconcile arbitrary finite-dimensional distributions and tail behavior for stationary IVP-based models. The linking mechanism between the marginals of a time series model and its asymptotic behavior turns out to be continuity, not stationarity or even additivity. We prove that any stationary finitely additive probability (charge) defined on cylinders has a stationary charge extension that can assign the divergence event any prescribed probability. Moreover, on the space of binary sequences, we consider IVP models that incorporate: (i) Stationarity. (ii) Continuity along $\mathscr{C}$. (iii) Almost sure support for divergence. (iv) Estimability of the divergence event from cylinders. (v) Nearly additive finite-dimensional distributions. We enhance the previous constructions of IVP's satisfying (i)-(iv) so that they satisfy (v) by agreeing with a stationary measure either exactly on one-dimensional cylinders or arbitrarily closely on a given class of bounded-dimensional cylinders. Our time series constructions follow from the observation that the algebra of cylinders and the tail $\sigma$-algebra are mutually nonsingular. We use the same idea to prove the existence of joints for general marginal IVP's. These constructions have implications for frequentist interpretations of probability.