Abstract
Our work is motivated by the study of empirical processes (such as flicker noise) that occur in stable systems yet give rise to observations with seemingly divergent time averages. Stationary models for such processes do not exist in the domain of numerical probability, as the ergodic theorems dictate the convergence of time averages of stationary and bounded processes. This has led us to investigate such models in the wider framework of interval-valued probability. In this paper we construct interval-valued probabilities on the space of infinite binary sequences that combine properties of (i) strict stationarity, (ii) unicity of extension from the algebra of cylinder sets to a wider collection containing salient asymptotic events, and (iii) almost sure support of divergence of time averages. These properties are not shared by conventional stochastic models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dempster, A. P. (1968). A generalization of Bayesian inference.J. Roy. Stat. Soc., Ser. B 30, 205–247.
Good, I. J. (1962). Subjective probability as the measure of a non-measurable set. In E. Nagel, P. Suppes, and A. Tarski (eds.),Logic, Methodology and Philosophy of Science, 319–329. Stanford University Press, Stanford.
Grize, Y. L. (1984). Towards a Stationary Continuous Lower Probability-Based Model for Flicker Noise, Ph.D. thesis, Cornell University.
Gize, Y. L., and Fine, T. L. (1987). Continuous lower probability-based models for stationary processes with bounded and divergent time averages.Ann. Prob. 15, 783–803.
Kroupa, V. (1983).Frequency Stability: Fundamentals and Measurement. IEEE Press, New York.
Kumar, A., and Fine, T. L. (1985). Stationary lower probabilities and unstable averages.Z. Wahrsch. verw. Gebiete 69, 1–17.
Papamarcou, A. (1987). Unstable Random Sequances as an Objective Basis for Interval-Valued Probability Models. Ph.D. thesis, Cornell University.
Papamarcou, A., and Fine, T. L. (1986). A Note on undominated lower probabilities.Ann. Prob. 14, 710–723.
Papamarcou, A., and Fine, T. L. (1991). Unstable collectives and envelopes of probability measures.Ann. Prob. 19 (2), (in press).
Shafer, G. (1976).A Mathematical Theory of Evidence. Princeton University Press, Princeton.
Smith, C. A. B. (1961). Consistency in statistical inference and decision.J. Roy. Stat. Soc., Ser. B 23, 1–25.
Walley, P. (1981). Coherent Lower and Upper Probabilities. Statistics Research Report, University of Warwick, Coventry, England.
Walley, P. (1990).Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London.
Walley, P., and Fine, T. L. (1982). Towards a frequentist theory of upper and lower probability.Ann. Stat. 10, 742–761.
Williams, P. M. (1976). Indeterminate probabilities. In M. Przelecki, K. Szaniawski, and R. Wojcicki (eds.),Formal Methods in the Methodology of Empirical Sciences. Reidel, Dordrecht, Holland.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Papamarcou, A., Fine, T.L. Stationarity and almost sure divergence of time averages in interval-valued probability. J Theor Probab 4, 239–260 (1991). https://doi.org/10.1007/BF01258736
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01258736