Continuous lower probability-based models for stationary processes with bounded and divergent time averages
type: Item
instance of: scholarly article
title: Continuous lower probability-based models for stationary processes with bounded and divergent time averages
zbMATH Open document ID: 0615.60035
author: Yves L. Grize
author: Terrence L. Fine
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: 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 \b{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 M of sets that includes the cylinder sets C and such that \b{P}(D\({}^*)>0\) where \(D^*=\{x=(x_ i)_{i\in Z}\in X: (1/n)\sum^{n-1}_{i=0}x_ i\) diverges as (n\(\to \infty)\}\). We show that these constraints are incompatible when \(M=2^ X\), but when \(M=C\) we are able to construct such a lower probability. Most of our results extend to the case of a compact marginal space \(X_ 0\).
Mathematics Subject Classification ID: 60G05
Mathematics Subject Classification ID: 60A05
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 3996762
zbMATH Keywords: empirical processes
zbMATH Keywords: characteristics of stationarity
zbMATH Keywords: finitely additive probability
zbMATH Keywords: modelling methodology
zbMATH Keywords: compact marginal space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992172
OpenAlex ID: W2042965659
DOI: 10.1214/AOP/1176992172
description: scientific article; zbMATH DE number 3996762