# A note on the law of large numbers in economics

## Author Info

Listed author(s):
• Patrizia Berti

()

(Dipartimento di Matematica Pura ed Applicata G. Vitali, Universita di Modena e Reggio-Emilia)

• Michele Gori

()

(Dipartimento di Matematica per le Decisioni, Universita di Firenze)

• Pietro Rigo

()

(Dipartimento di Economia Politica e Metodi Quantitativi, Universita di Pavia)

## Abstract

Let $(S,\mathcal{B},\Gamma)$ and $(T,\mathcal{C},Q)$ be probability spaces, with $Q$ nonatomic, and $\mathcal{H}=\{H\in\mathcal{C}:Q(H)>0\}$. In some economic models, the following conditional law of large numbers (LLN) is requested. There are a probability space $(\Omega,\mathcal{A},P)$ and a process $X=\{X_t:t\in T\}$, with state space $(S,\mathcal{B})$, satisfying \begin{gather*} \text{for each }H\in\mathcal{H},\text{ there is }A_H\in\mathcal{A}\text{ with }P(A_H)=1\text{ such that } \\t\mapsto X(t,\omega)\text{ is measurable and }\,Q\bigl(\{t:X(t,\omega)\in\cdot\}\mid H\bigr)=\Gamma(\cdot)\,\text{ for }\omega\in A_H. \end{gather*} If $\Gamma$ is not trivial and the $\sigma$-field $\mathcal{C}$ countably generated, the conditional LLN fails in the usual (countably additive) setting. Instead, as shown in this note, it holds in a finitely additive setting. Also, $X$ can be taken to have any given distribution. In fact, for any consistent set $\mathcal{P}$ of finite dimensional distributions, there are a finitely additive probability space $(\Omega,\mathcal{A},P)$ and a process $X$ such that $X\sim\mathcal{P}$ and the conditional LLN is satisfied.

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## Bibliographic Info

Paper provided by Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa in its series Working Papers - Mathematical Economics with number 2009-10.

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 Length: 8 Date of creation: Dec 2009 Date of revision: Nov 2010 Handle: RePEc:flo:wpaper:2009-10 Contact details of provider: Postal: Via delle Pandette 9 50127 - Firenze - ItalyPhone: +39 055 2759707Fax: +39 055 2759913Web page: http://www.disei.unifi.it/Email: More information through EDIRC

## References

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1. Judd, Kenneth L., 1985. "The law of large numbers with a continuum of IID random variables," Journal of Economic Theory, Elsevier, vol. 35(1), pages 19-25, February.
2. Sun, Yeneng, 2006. "The exact law of large numbers via Fubini extension and characterization of insurable risks," Journal of Economic Theory, Elsevier, vol. 126(1), pages 31-69, January.
3. Al-Najjar, Nabil I., 2004. "Aggregation and the law of large numbers in large economies," Games and Economic Behavior, Elsevier, vol. 47(1), pages 1-35, April.
4. Harald Uhlig, 1996. "A law of large numbers for large economies (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 41-50.
5. Sun, Yeneng & Zhang, Yongchao, 2009. "Individual risk and Lebesgue extension without aggregate uncertainty," Journal of Economic Theory, Elsevier, vol. 144(1), pages 432-443, January.
6. Al-Najjar, Nabil I., 2008. "Large games and the law of large numbers," Games and Economic Behavior, Elsevier, vol. 64(1), pages 1-34, September.
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