Existence of independent random matching

This paper shows the existence of independent random matching of a large (continuum) population in both static and dynamic systems, which has been popular in the economics and genetics literatures. We construct a joint agent-probability space, and randomized mutation, partial matching and match-induced type-changing functions that satisfy appropriate independence conditions. The proofs are achieved via nonstandard analysis. The proof for the dynamic setting relies on a new Fubini-type theorem for an infinite product of Loeb transition probabilities, based on which a continuum of independent Markov chains is derived from random mutation, random partial matching and random type changing.Comment: Published at http://dx.doi.org/10.1214/105051606000000673 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mixing and double recurrence in probability groups

We define a class of groups equipped with an invariant probability measure, which includes all compact groups and is closed under taking ultraproducts with the induced Loeb measure. We call these probability groups and develop the basics of the theory of their measure-preserving actions on probability spaces, including a natural notion of mixing. A short proof reveals that for probability groups mixing implies double recurrence, which generalizes a theorem of Bergelson and Tao proved for ultraproducts of finite groups. Moreover, a quantitative version of our proof gives that $\epsilon$-approximate mixing implies $3\sqrt{\epsilon}$-approximate double recurrence. Examples of approximately mixing probability groups are quasirandom groups introduced by Gowers, so the last theorem generalizes and sharpens the corresponding results for quasirandom groups of Bergelson and Tao, as well as of Austin.Comment: Corrected the definitions of probability groups and their actions. Added a quick overview of ultraproducts and their countable-compactness, as well as the Loeb measure constructio

Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds

We prove a general form of the regularity theorem for uniformity norms, and deduce a generalization of the Green-Tao-Ziegler inverse theorem, extending it to a class of compact nilspaces including all compact abelian groups and nilmanifolds. We derive these results from a structure theorem for cubic couplings, thereby unifying these results with the ergodic structure theorem of Host and Kra. The proofs also involve new results on nilspaces. In particular, we obtain a new stability result for nilspace morphisms. We also strengthen a result of Gutman, Manners and Varju, by proving that a k-step compact nilspace of finite rank is a toral nilspace (in particular, a connected nilmanifold) if and only if its k-dimensional cube set is connected. We also prove that if a morphism from a cyclic group of prime order into a compact finite-rank nilspace is sufficiently balanced (a quantitative form of multidimensional equidistribution), then the nilspace is toral.Comment: 35 page

Ultraproducts and metastability

Given a convergence theorem in analysis, under very general conditions a model-theoretic compactness argument implies that there is a uniform bound on the rate of metastability. We illustrate with three examples from ergodic theory

Perfect Competition

In his 1987 entry on ‘Perfect Competition’ in The New Palgrave, the author reviewed the question of the perfectness of perfect competition, and gave four alternative formalisations rooted in the so-called Arrow-Debreu-Mckenzie model. That entry is now updated for the second edition to include work done on the subject during the last twenty years. A fresh assessment of this literature is offered, one that emphasises the independence assumption whereby individual agents are not related except through the price system. And it highlights a ‘linguistic turn’ whereby Hayek’s two fundamental papers on ‘division of knowledge’ are seen to have devastating consequences for this research programme.Allocation of Resources; Perfect Competition; Exchange Economy