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A note on the law of large numbers in economics

Author

Listed:
  • 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.

Suggested Citation

  • Patrizia Berti & Michele Gori & Pietro Rigo, 2009. "A note on the law of large numbers in economics," Working Papers - Mathematical Economics 2009-10, Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa, revised Nov 2010.
  • Handle: RePEc:flo:wpaper:2009-10
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    References listed on IDEAS

    as
    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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    More about this item

    Keywords

    Aggregate uncertainty; Extension; Finitely additive probability; Individual risk; Law of large numbers.;

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • D80 - Microeconomics - - Information, Knowledge, and Uncertainty - - - General

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