Substituting Independent Processes

TL;DR

This paper constructs a process with uniform marginals that preserves mixing properties and is a finitary factor of a given process, extending empirical distribution approximation results to more general cases.

Contribution

It introduces a method to represent stationary processes as finitary factors of processes with uniform marginals, preserving order and mixing properties.

Findings

Existence of a process U with uniform marginals and same mixing as X

X is a finitary factor of U with coding length 1

Extension of empirical distribution approximation to general weakly dependent vectors

Abstract

It is shown by constructing Rohlins canonical measures that for a strictly stationary, d-dimensional vector-valued process X there exists another strictly stationary d-dimensional process U with uniform one-dimensional marginals and with the same mixing properties as X, such that X is a finitary factor of U of coding length 1, and such that the projection map is order preserving in each coordinate. As an application this extends the a.s. approximation of the empirical distribution function of weakly dependent random vectors with continuous distribution function in [1] and [3] to the general case.