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The Monte Carlo integral of a continuum of independent random variables
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Hammond, Peter J. (2023) The Monte Carlo integral of a continuum of independent random variables. Working Paper. Coventry, UK: University of Warwick. Department of Economics. Warwick economics research papers series (WERPS), 1479 . (Unpublished)
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Official URL: https://warwick.ac.uk/fac/soc/economics/research/w...
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.
Item Type: | Working or Discussion Paper (Working Paper) | ||||||
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Alternative Title: | |||||||
Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Social Sciences > Economics | ||||||
Library of Congress Subject Headings (LCSH): | Monte Carlo method, Random variables, Numerical integration, Continuity | ||||||
Series Name: | Warwick economics research papers series (WERPS) | ||||||
Publisher: | University of Warwick. Department of Economics | ||||||
Place of Publication: | Coventry, UK | ||||||
ISSN: | 2059-4283 | ||||||
Official Date: | October 2023 | ||||||
Dates: |
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Volume: | 1479 | ||||||
Institution: | University of Warwick | ||||||
Status: | Not Peer Reviewed | ||||||
Publication Status: | Unpublished | ||||||
Access rights to Published version: | Free Access (unspecified licence, 'bronze OA') | ||||||
Description: | This paper also appears as CRETA Discussion paper 80 |
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Open Access Version: |
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