IDEAS home Printed from https://ideas.repec.org/p/wrk/wcreta/80.html
   My bibliography  Save this paper

The Monte Carlo Integral of a Continuum of Independent Random Variables

Author

Listed:
  • Hammond, Peter J.

    (Department of Economics and CRETA, University of Warwick)

Abstract

Consider a continuum of independent and identically distributed random variables corresponding to the points of the unit interval [0; 1]. Known technical diffculties are complemented by showing directly that the random sample path is almost surely not a Lebesgue measurable function. This refutes the common claim that, because of some version of the law of large numbers, the integral of each sample path equals the common mean of each random variable. To obtain a valid and useful result, we apply to the continuum of random variables the Monte Carlo method of numerical integration based on limits as the sample size tends to infinity of empirical finite sample averages of the realized random values. The resulting Monte Carlo integral is almost surely a degenerate random variable concentrated on the mean. A suitably modified version works when the different indexed random variables are merely independent with cumulative distribution functions that are measurable w.r.t. the index. Further generalizations to Monte Carlo integrals of conditionally independent random variables result, under conditions discussed in Hammond and Sun (2008, 2021), in non-degenerate random integrals that are measurable w.r.t. the conditioning -algebra.

Suggested Citation

  • Hammond, Peter J., 2023. "The Monte Carlo Integral of a Continuum of Independent Random Variables," CRETA Online Discussion Paper Series 80, Centre for Research in Economic Theory and its Applications CRETA.
    • Handle: RePEc:wrk:wcreta:80
    as

    Download full text from publisher

    File URL: https://warwick.ac.uk/fac/soc/economics/research/centres/creta/papers/manage/creta80_-_peter_hammond.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Sun, Yeneng, 1998. "A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN1," Journal of Mathematical Economics, Elsevier, vol. 29(4), pages 419-503, May.
    2. Kloek, Tuen & van Dijk, Herman K, 1978. "Bayesian Estimates of Equation System Parameters: An Application of Integration by Monte Carlo," Econometrica, Econometric Society, vol. 46(1), pages 1-19, January.
    3. José Jorge & Joana Rocha, 2015. "A Primer On Global Games Applied To Macroeconomics And Finance," Journal of Economic Surveys, Wiley Blackwell, vol. 29(5), pages 869-886, December.
    4. Martin Jensen, 2010. "Aggregative games and best-reply potentials," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 43(1), pages 45-66, April.
    5. van Dijk, H. K. & Kloek, T., 1980. "Further experience in Bayesian analysis using Monte Carlo integration," Journal of Econometrics, Elsevier, vol. 14(3), pages 307-328, December.
    6. Feldman, Mark & Gilles, Christian, 1985. "An expository note on individual risk without aggregate uncertainty," Journal of Economic Theory, Elsevier, vol. 35(1), pages 26-32, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. James A. Chalfant & Richard S. Gray & Kenneth J. White, 1991. "Evaluating Prior Beliefs in a Demand System: The Case of Meat Demand in Canada," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 73(2), pages 476-490.
    2. Khan, M. Ali & Sun, Yeneng, 2001. "Asymptotic Arbitrage and the APT with or without Measure-Theoretic Structures," Journal of Economic Theory, Elsevier, vol. 101(1), pages 222-251, November.
    3. Ardia, David & Hoogerheide, Lennart F., 2010. "Efficient Bayesian estimation and combination of GARCH-type models," MPRA Paper 22919, University Library of Munich, Germany.
    4. Sahm, Marco, 2006. "Essays in Public Economic Theory," Munich Dissertations in Economics 5633, University of Munich, Department of Economics.
    5. Steel, Mark F. J., 1991. "A Bayesian analysis of simultaneous equation models by combining recursive analytical and numerical approaches," Journal of Econometrics, Elsevier, vol. 48(1-2), pages 83-117.
    6. VAN DIJK, Herman K., 1987. "Some advances in Bayesian estimations methods using Monte Carlo Integration," LIDAM Reprints CORE 783, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Khan, M. Ali & Sun, Yeneng, 2003. "Exact arbitrage, well-diversified portfolios and asset pricing in large markets," Journal of Economic Theory, Elsevier, vol. 110(2), pages 337-373, June.
    8. Lennart F. Hoogerheide & Johan F. Kaashoek, 2004. "Functional Approximations to Likelihoods/Posterior Densities: A Neural Network Approach to Efficient Sampling," Computing in Economics and Finance 2004 74, Society for Computational Economics.
    9. Konrad Podczeck, 2010. "On existence of rich Fubini extensions," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 45(1), pages 1-22, October.
    10. van Dijk, H. K. & Kloek, T., 1983. "Experiments With Some Alternatives For Simple Importance Sampling In Monte Carlo Integration," Econometric Institute Archives 272281, Erasmus University Rotterdam.
    11. van Dijk, H. K. & Kloek, T., 1982. "Monte Carlo Analysis Of Skew Posterior Distributions: An Illustrative Econometric Example," Econometric Institute Archives 272268, Erasmus University Rotterdam.
    12. Heckelei, Thomas & Mittelhammer, Ronald C., 1996. "Bayesian Bootstrap Analysis of Systems of Equations," Discussion Papers 18786, University of Bonn, Institute for Food and Resource Economics.
    13. Karavaev, Andrei, 2008. "A Theory of Continuum Economies with Idiosyncratic Shocks and Random Matchings," MPRA Paper 7445, University Library of Munich, Germany.
    14. Guyonne Kalb & Jenny Williams, 2003. "Delinquency and gender," Applied Economics Letters, Taylor & Francis Journals, vol. 10(7), pages 425-429.
    15. Hoogerheide, L.F. & Kaashoek, J.F. & van Dijk, H.K., 2003. "Neural network approximations to posterior densities: an analytical approach," Econometric Institute Research Papers EI 2003-38, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    16. Hoogerheide, Lennart & van Dijk, Herman K., 2010. "Bayesian forecasting of Value at Risk and Expected Shortfall using adaptive importance sampling," International Journal of Forecasting, Elsevier, vol. 26(2), pages 231-247, April.
    17. Lennart Hoogerheide & Anne Opschoor & Herman K. van Dijk, 2011. "A Class of Adaptive EM-based Importance Sampling Algorithms for Efficient and Robust Posterior and Predictive Simulation," Tinbergen Institute Discussion Papers 11-004/4, Tinbergen Institute.
    18. Jerez, Belen, 2003. "A dual characterization of incentive efficiency," Journal of Economic Theory, Elsevier, vol. 112(1), pages 1-34, September.
    19. Van Dijk, Herman K. & Kloek, Teun & Boender, C. Guus E., 1985. "Posterior moments computed by mixed integration," Journal of Econometrics, Elsevier, vol. 29(1-2), pages 3-18.
    20. Denis Fougère & Thierry Kamionka, 2003. "Bayesian inference for the mover-stayer model in continuous time with an application to labour market transition data," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 18(6), pages 697-723.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wrk:wcreta:80. See general information about how to correct material in RePEc.

    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 CitEc recognized a bibliographic reference but did not link an item in RePEc 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 RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Margaret Nash (email available below). General contact details of provider: https://edirc.repec.org/data/dewaruk.html .

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.