Summary
LetX(i),iε[0; 1] be a collection of identically distributed and pairwise uncorrelated random variables with common finite meanμ and variance σ2. This paper shows the law of large numbers, i.e. the fact that ∝ 10 X(i)di=μ. It does so by interpreting the integral as a Pettis-integral. Studying Riemann sums, the paper first provides a simple proof involving no more than the calculation of variances, and demonstrates, that the measurability problem pointed out by Judd (1985) is avoided by requiring convergence in mean square rather than convergence almost everywhere. We raise the issue of when a random continuum economy is a good abstraction for a large finite economy and give an example in which it is not.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Al-Najjar, N. I.: Decomposition and characterization of risk with a continuum of random variables. Econometrica63, 1195–1224 (1995)
Bewley, T.: Appendix to Stationary monetary equilibrium with a continuum of independently fluctuating consumers. In: Hildenbrand, Mas-Colell (eds.) Contributions to mathematical economics in Honor of G. Debreu, Amsterdam: North-Holland, 1986
Diestel, J., Uhl, J. J. Jr.: Vector measures. Mathematical surveys # 15. Providence, R. I.: American Mathematical Society, 1977
Diamond, D. W., Dybvig, P. H.: Bandk runs, deposit insurance and liquidity. Journal of Political Economy91, 401–419 (1983)
Green, E.: Lending and the smoothing of uninsurable income. In: E. C. Prescott, and N. Wallace (eds.) Contractual arrangements for intertemporal trade. Minnesota studies in macroeconomis, Vol. 1, pp. 3–25. Minneapolis: University of Minnesota Press, 1987
Hildenbrand, W.: Core and equilibria of a large economy. Princeton: Princeton University Press, 1974
Judd, K. L.: The law of large numbers with a continuum of IID random variables. Journal of Economic Theory35, 19–25 (1985)
Lucas, R. E.: Equilibrium in a pure currency economy. Economic Inquiry18, 203–220 (1980)
Mas-Colell, A.: The theory of general equilibrium: A differentiable approach. Cambridge: Cambridge University Press, 1985
Mas-Colell, A., Vives, X.: Implementation in economies with a continuum of agents. Review of Economic Studies60, 613–629 (1993)
Prescott, E. C., Townsend, R. M.: Pareto optima and competitive equilibria with adverse selection and moral hazard. Econometrica52, 21–45 (1984)
Pettis, B. J.: On integration in vector spaces. Trans. Amer. Math. Soc.44, 277–304 (1938)
Author information
Authors and Affiliations
Additional information
I am indebted to Hugo Hopenhayn. Furthermore I would like to thank Dilip Abreu, Glenn Donaldson, Ed Green, Ramon Marimon, Nabil Al-Najjar, Victor Rios-Rull, Timothy van Zandt and the editor for useful comments. The first version of this paper was written in 1987.
Rights and permissions
About this article
Cite this article
Uhlig, H. A law of large numbers for large economies. Econ Theory 8, 41–50 (1996). https://doi.org/10.1007/BF01212011
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01212011