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Continuum of Random Variables

  • (with Yeneng Sun) “Monte Carlo Simulation of Macroeconomic Risk with a Continuum of Agents: The Symmetric Case,” Economic Theory 21 (2003), 743–766; also in C.D. Aliprantis et al. (eds.) Assets, Beliefs, and Equilibria in Economic Dynamics: Essays in Honor of Mordecai Kurz (Berlin: Springer-Verlag, 2003), pp. 709–732.


    Suppose a large economy with individual risk is modeled by a continuum of pairwise exchangeable random variables (i.i.d., in particular). Then the relevant stochastic process is jointly measurable only in degenerate cases. Yet in Monte Carlo simulation, the average of a large finite draw of the random variables converges almost surely. Several necessary and sufficient conditions for such “Monte Carlo convergence” are given. Also, conditioned on the associated Monte Carlo σ-algebra, which represents macroeconomic risk, individual agents’ random shocks are independent. Furthermore, a converse to one version of the classical law of large numbers is proved. PDF file of preprint: Springer link

  • (with Yeneng Sun) “Joint Measurability and the One-way Fubini Property for a Continuum of Independent Random Variables,” Proceedings of the American Mathematical Society 134 (2006), 737–747.


    As is well known, a continuous parameter process with mutually independent random variables is not jointly measurable in the usual sense. This paper proposes using a natural “one-way Fubini” property that guarantees a unique meaningful solution to this joint measurability problem when the random variables are independent even in a very weak sense. In particular, if F is the smallest extension of the usual product sigma-algebra such that the process is measurable, then there is a unique probability measure ν on F such that the integral of any ν-integrable function is equal to a double integral evaluated in one particular order. Moreover, in general this measure cannot be further extended to satisfy a two-way Fubini property. However, the extended framework with the one-way Fubini property not only shares many desirable features previously demonstrated under the stronger two-way Fubini property, but also leads to a new characterization of the most basic probabilistic concept — stochastic independence in terms of regular conditional distributions. PDF file of preprint: AMS link

  • (with Yeneng Sun) “The Essential Equivalence of Pairwise and Mutual Conditional Independence,” Probability Theory and Related Fields 135 (2006), 415–427.


    For a large collection of random variables, pairwise conditional independence and mutual conditional independence are shown to be essentially equivalent. Unlike in the finite setting, a large collection of random variables remains essentially conditionally independent under further conditioning. The essential equivalence of pairwise and multiple versions of exchangeability also follows as a corollary. Our proof is based on an iterative extension of Bledsoe and Morse’s completion of a product measure on a pair of measure spaces. PDF file of preprint

  • (with Yeneng Sun) “Monte Carlo Simulation of Macroeconomic Risk with a Continuum of Agents: The General Case,” Economic Theory DOI 10.1007/s00199-007-0279-7 (published online, 7 September 2007).

    In large random economies with heterogeneous agents, a standard stochastic framework presumes a random macro state, combined with idiosyncratic micro shocks. This can be formally represented by a random process consisting of a continuum of random variables that are conditionally independent given the macro state. However, this process satisfies a standard joint measurability condition only if there is essentially no idiosyncratic risk at all. Based on iteratively complete product measure spaces, we characterize the validity of the standard stochastic framework via Monte Carlo simulation as well as event-wise measurable conditional probabilities. These general characterizations also allow us to strengthen some earlier results related to exchangeability and independence. PDF file of Warwick Economics Research Paper: Springer link

  • (with Yeneng Sun) “Characterization of Risk: A Sharp Law of Large Numbers,” Warwick Economic Research Paper, no. 806 (2007).


    An extensive literature in economics uses a continuum of random variables to model individual random shocks imposed on a large population. Let H denote the Hilbert space of square-integrable random variables. A key concern is to characterize the family of all H-valued functions that satisfy the law of large numbers when a large sample of agents is drawn at random. We use the iterative extension of an infinite product measure introduced in [6] to formulate a “sharp” law of large numbers. We prove that an H-valued function satisfies this law if and only if it is both Pettis-integrable and norm integrably bounded. PDF file