Foundations for Random Matching
This appendix summarizes conditions from Duffie and Sun (2011) for an exact law of large numbers for random matching of a [continuum] of investors in a static setting. Existence of models satisfying these conditions is shown in Duffie and Sun (2007). These sources also provide Markovian dynamics for agents' type distributions in multiperiod settings.
Let (Ω, F,ℙ) be a probability space and let (G, G, γ) be an atomless space of agents of total mass γ(G) = 1. For example, G could be the unit interval [0,1] with the uniform distribution (Lebesgue measure). Let (G × Ω, G ⊗ F, , γ ⊗ ℙ) be the usual product space. For a function f on G × Ω. and for some (i, ω) ∈ G × Ω, a function fi = f(i,·) represents the outcome for agent i across the different states, and the function fω= f(·, ω) on G represents the cross section of outcomes in state ω for the different agents.
In order to work with independent type processes arising from random matching, we define an extension of the usual measure-theoretic product that retains the Fubini property. A formal definition is as follows.
Definition 1A probability space (G × Ω, W, Q) extending the usual product space (G × Ω., G ⊗ F, γ ⊗ ℙ) is said to be a Fubini extension if, for any real-valued Q-integrable function g on (G × Ω, W), the functions gi= g(i,·) and gω, = g(·,ω) are integrable, respectively, on (Ω., F, ℙ) for γ-almost all i ∈ i and on (G, G, γ) for P-almost all ω ∈ Ω; and if, moreover,and are integrable, respectively, on (G, G,γ) and on (Ω, F, ℙ)with .