IDEAS home Printed from
MyIDEAS: Log in (now much improved!) to save this paper

A note on the law of large numbers in economics

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)

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.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL:
Download Restriction: no

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.

in new window

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 - Italy

Phone: +39 055 2759707
Fax: +39 055 2759913
Web page:

More information through EDIRC

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

in new window

  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.
Full references (including those not matched with items on IDEAS)

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

When requesting a correction, please mention this item's handle: RePEc:flo:wpaper:2009-10. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Michele Gori)

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.