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Hyperfinite Law of Large Numbers

Published online by Cambridge University Press:  15 January 2014

Yeneng Sun
Yeneng Sun*
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 0511 E-mail: [email protected]
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Abstract

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.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 2 , Issue 2 , June 1996 , pp. 189 - 198
Copyright
Copyright © Association for Symbolic Logic 1986

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References

REFERENCES

[1] Anderson, R. M., Non-standard analysis with applications to economics, Handbook of mathematical economics (Hildebrand, W. and Sonnenschein, H., editors), vol. IV, North-Holland, New York, 1991.Google Scholar
[2] Bewley, T. F., Stationary monetary equilibrium with a continuum of independently fluctuating consumers, Contributions to mathematical economics: in honor of Gerard Debreu (Hildebrand, W. and Mas-Colell, A., editors), North-Holland, New York, 1986.Google Scholar
[3] Doob, J. L., Stochastic processes, Wiley, New York, 1953.Google Scholar
[4] Henson, C. W. and Keisler, H. J., On the strength of nonstandard analysis, The Journal of Symbolic Logic, vol. 51 (1986), pp. 377–386.Google Scholar
[5] Hurd, A. E. and Loeb, P. A., An introduction to nonstandard real analysis, Academic Press, Orlando, Florida, 1985.Google Scholar
[6] Judd, K. L., The law of large numbers with a continuum of iid random variables, Journal of Economic Theory, vol. 35 (1985), pp. 19–25.CrossRefGoogle Scholar
[7] Keisler, H. J., Infinitesimals in probability theory, Nonstandard analysis and its applications (Cutland, N. J., editor), Cambridge University Press, Cambridge, 1988.Google Scholar
[8] Loeb, P. A., Conversion from nonstandard to standard measure spaces and applications in probability theory, Transactions of the American Mathematical Society, vol. 211 (1975), pp. 113–122.Google Scholar
[9] Loéve, M., Probability theory II, 4th ed., Springer-Verlag, New York, 1977.Google Scholar
[10] Lucas, R. E. and Prescott, E. C., Equilibrium search and unemployment, Journal of Economic Theory, vol. 7 (1974), pp. 188–209.CrossRefGoogle Scholar
[11] Sun, Y.N., Distributional properties of correspondences on Loeb spaces, to appear in Journal of Functional Analysis .Google Scholar
[12] Sun, Y.N., The law of large numbers for set-valued processes and stationary equilibria, National University of Singapore, preprint.Google Scholar
[13] Sun, Y.N., A theory of hyperfinite processes, National University of Singapore, preprint.Google Scholar
[14] Sun, Y.N., Vector valued processes and the law of large numbers, to appear.Google Scholar