Summary
This paper explores the possibilities for probability-like models of stationary nondeterministic phenomena that possess divergent but bounded time averages. A random sequence described by a stationary probability measure must have almost surely convergent time averages whenever it has almost surely bounded time averages. Hence, no measure can provide the mathematical model we desire. In turning to lower probability based models we first explore the relationships between divergence, stationarity, and monotone continuity and those between monotone continuity and unicity of extensions. We then construct several examples of stationary lower probabilities for sequences of uniformly bounded random variables such that divergence of time averages occurs with lower probability one. We conclude with some remarks on the problem of estimating lower probability models on the basis of cylinder set observations.
Article PDF
Similar content being viewed by others
References
Good, I.J.: Subjective probability as the measure of a non-measurable set. In: Nagel, E., Suppes, P., Tarski, A. (eds.), Logic, methodology and philosophy of science. Stanford: University Press 1962
Grize, Y.L.: Towards a stationary continuous lower probability-based model for Flicker noise. Ph. D. dissertation. N.Y.: Cornell University, Ithaca 1984
Huber, P.J.: Kapazitäten statt Wahrscheinlichkeiten? Gedanken zur Grundlegung der Statistik. Jber. d. Dt. Math.-Verein 78, 81–92 (1976)
Kalikow, S.: Private communication (1984)
Kroupa, V.: Frequency stability: Fundamentals and measurement. New York: Inst. of Electric and Electrical Engineering 1983
Kumar, A.: Lower Probabilities on Infinite Spaces and Instability of Stationary Sequences. Ph.D. dissertation. N.Y.: Cornell University, Ithaca 1982
Neveu, J.: Mathematical foundations of the calculus of probability. San Francisco: Holden Day 1965
Papamarcou, A.: Some results on undominated lower probabilities. M.S. dissertation, Cornell University, Ithaca, N.Y. 1983
Royden, H.L.: Real analysis. New York: Macmillan 1968
Shafer, G.: A mathematical theory of evidence. Princeton: Princeton University Press 1976
Smith, C.A.B.: Consistency in statistical inference and decision. J. Royal Statist. Soc., Ser. B 23, 1–25 (1961)
Suppes, P.: The measurement of belief. J. Royal Stat. Soc., Ser. B 36, 160–175 (1974)
Walley, P.: Coherent lower (and upper) probabilities. Technical report. Department of Statistics, University of Warwick, England (1981)
Walley, P., Fine, T.L.: Towards a frequentist theory of upper and lower probability. Annal. Statist. 10, 741–761 (1982)
Williams, P.M.: Indeterminate probabilities, In: M. Przelecki, K. Szaniawski, R. Wojcicki (eds.). Formal methods in the methodology of empirical sciences. Dordrecht: Reidel 1976
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kumar, A., Fine, T.L. Stationary lower probabilities and unstable averages. Z. Wahrscheinlichkeitstheorie verw Gebiete 69, 1–17 (1985). https://doi.org/10.1007/BF00532581
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00532581