Summary
Let ℳ denote the class of infinite product probability measures μ=μ 1×μ 2×⋯ defined on an infinite product of replications of a given measurable space (X, A), and let ℋ denote the subset of ℳ for which μ(A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member μ, of ℳ belongs to ℋ. In the present paper, the class ℋ is shown to possess several closure properties. E.g., if μ∈ℋ and μ 0≪μ n for some n ≧1, then μ 0×μ 1×μ 2×...∈ℋ. While the current results do not permit a complete characterization of ℋ they demonstrate conclusively that ℋ is a much larger subset of ℳ than previous results indicated. The interesting special case X={0,1} is discussed in detail.
Article PDF
Similar content being viewed by others
References
Aldous, D., Pitman, J.: On the zero-one law for exchangeable events. Unpublished manuscript (1977)
Blum, J.R., Pathak, P.K.: A note on the zero-one law. Ann. Math. Statist. 45, 1008–1009 (1972)
Hewitt, E., Savage, L.J.: Symmetric measures on Cartesian products. Trans. Amer. Math. Soc., 80, 470–501 (1955)
Horn, S., Schach, S.: An extension of the Hewitt-Savage zero-one law. Ann. Math. Statist. 41, 2130–2131 (1970)
Neveu, J.: Discrete-Parameter Martingales. Amsterdam: North-Holland/American Elsevier 1975
Sendler, W.: A note on the proof of the zero-one law of Blum and Pathak. Ann. Probability, 3, 1055–1058 (1975)
Author information
Authors and Affiliations
Additional information
Research supported by the National Science Foundation under grant No. MCS75-07556
Rights and permissions
About this article
Cite this article
Simons, G. Some extensions of the Hewitt-Savage zero-one law. Z. Wahrscheinlichkeitstheorie verw Gebiete 42, 167–173 (1978). https://doi.org/10.1007/BF00536052
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00536052