[Paper Review] Individual-level randomness in a nonatomic population

Edward J. Green|arXiv (Cornell University)|Apr 1, 2019

Economic theories and models7 references41 citations

TL;DR

The paper constructs an uncountable family of i.i.d. random vectors indexed by a nonatomic measure space, achieving an exact Glivenko-Cantelli analogue and homogeneous sampling properties for economic applications.

ABSTRACT

This paper provides a construction of an uncountable family of i.i.d. random vectors, indexed by the points of a nonatomic measure space, such that (a) a sample is a measurable function from the index space, and (b) an idealization of the Glivenko-Cantelli theorem holds exactly with respect to the measure on that space. That is, samples possess a.s. the distribution from which they are drawn. Moreover, any subspace of the index space with positive measure inherits the same property. This homogeneity property is important for applications of the construction in economics.

Motivation & Objective

• Motivate the need for a secure foundation for models with idiosyncratic risks in large economies.
• Provide a set-theoretic construction of an uncountable i.i.d. family indexed by a nonatomic probability space.
• Ensure that samples preserve the population distribution and yield a Glivenko-Cantelli-like property exactly with respect to the population measure.
• Demonstrate a homogeneity property: every positive-measure subspace inherits the same sampling behavior.

Proposed method

• Use a set-theoretic Kolmogorov-type construction to define an i.i.d. family of random vectors with a specified distribution.
• Index random vectors by elements of an abstract nonatomic probability space rather than by unit interval points.
• Construct a rich function to generate a homogeneous sample-distribution limit through a dual Kolmogorov extension framework.
• Establish a sample-distribution limit that satisfies an exact analogue of the Glivenko-Cantelli property relative to the population measure.

Experimental results

Research questions

• RQ1 Can one construct an uncountable, homogeneous family of i.i.d. random vectors indexed by a nonatomic space whose samples are measurable functions from the index space?
• RQ2 Is it possible to obtain an exact analogue of the Glivenko-Cantelli theorem in this nonatomic, continuum setting?
• RQ3 Does every positive-measure subspace inherit the same sampling property under normalization to a probability measure?

Key findings

• An uncountable family of i.i.d. random vectors indexed by a nonatomic space exists with the desired exact sampling property.
• The construction yields samples that a.s. have the same distribution as the original i.i.d. vectors, relative to the index-space measure.
• Any subspace of the index space with positive measure, when normalized to a probability measure, retains the same sampling property.
• A homogeneous sample-distribution limit is achieved, satisfying a version of the Glivenko-Cantelli idealization for measurable subsets of the population.
• Under mild conditions, the constructed homogeneous limit is not jointly measurable in its two arguments (sample and population), reflecting a fundamental non-measurability feature of such limit objects.