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.
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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.
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Uhlig, H. A law of large numbers for large economies. Econ Theory 8, 41–50 (1996). https://doi.org/10.1007/BF01212011
- Economic Theory
- Simple Proof
- Measurability Problem
- Large Economy
- Uncorrelated Random Variable