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Hyperfinite Law of Large Numbers

  • Yeneng Sun (a1)

The Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods. An asymptotic interpretation of results established here shows that for a triangular array (or a sequence) of random variables, asymptotic uncorrelatedness or asymptotic pairwise independence is necessary and sufficient for the validity of appropriate versions of the law of large numbers. Our intrinsic characterization of almost sure pairwise independence leads to the equivalence of various multiplicative properties of random variables.

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[1] R. M. Anderson , Non-standard analysis with applications to economics, Handbook of mathematical economics ( W. Hildebrand and H. Sonnenschein , editors), vol. IV, North-Holland, New York, 1991.

[4] C. W. Henson and H. J. Keisler , On the strength of nonstandard analysis, The Journal of Symbolic Logic, vol. 51 (1986), pp. 377386.

[6] K. L. Judd , The law of large numbers with a continuum of iid random variables, Journal of Economic Theory, vol. 35 (1985), pp. 1925.

[7] H. J. Keisler , Infinitesimals in probability theory, Nonstandard analysis and its applications ( N. J. Cutland , editor), Cambridge University Press, Cambridge, 1988.

[8] P. A. Loeb , Conversion from nonstandard to standard measure spaces and applications in probability theory, Transactions of the American Mathematical Society, vol. 211 (1975), pp. 113122.

[10] R. E. Lucas and E. C. Prescott , Equilibrium search and unemployment, Journal of Economic Theory, vol. 7 (1974), pp. 188209.

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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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