CHAPTER 10 - REPRESENTATIONS AND COUPLINGS

Summary

1. SECTION 1 illustrates the usefulness of coupling, by means of three simple examples.

2. SECTION 2 describes how sequences of random elements of separable metric spaces that converge in distribution can be represented by sequences that converge almost surely.

3. SECTION *3 establishes Strassen's Theorem, which translates the Prohorov distance between two probability measures into a coupling.

4. SECTION *4 establishes Yurinskii's coupling for sums of independent random vectors to normally distributed random vectors.

5. SECTION 5 describes a deceptively simple example (Tusnády's Lemma) of a quantile coupling, between a symmetric Binomial distribution and its corresponding normal approximation.

6. SECTION 6 uses the Tusnády Lemma to couple the Haar coefficients for the expansions of an empirical process and a generalized Brownian Bridge.

7. SECTION 7 derives one of most striking results of modern probability theory, the KMT coupling of the uniform empirial process with the Brownian Bridge process.

What is coupling?

A coupling of two probability measures, P and Q, consists of a probability space (Ω, ℱ, ℙ) supporting two random elements X and F, such that X has distribution P and Y has distribution Q. Sometimes interesting relationships between P and Q can be coded in some simple way into the joint distribution for X and Y. Three examples should make the concept clearer.

Example. Let P_α denote the Bin(n,α) distribution. As α gets larger, the distribution should “concentrate on bigger values.” More precisely, for each fixed x, the tail probability P_α[x, n] should be an increasing function of α. A coupling argument will give an easy proof.