Concentration of measure without independence: a unified approach via the martingale method

TL;DR

This paper extends the martingale method to derive concentration inequalities for dependent random variables using Wasserstein matrices, unifying and strengthening results beyond the independent case.

Contribution

It introduces Wasserstein matrices as a formalism to analyze concentration without independence, unifying and enhancing existing inequalities via the martingale approach.

Findings

Recovered and sharpened concentration inequalities for nonproduct measures.

Demonstrated the power of Wasserstein matrices in the martingale framework.

Unified approach applicable to dependent random variables.

Abstract

The concentration of measure phenomenon may be summarized as follows: a function of many weakly dependent random variables that is not too sensitive to any of its individual arguments will tend to take values very close to its expectation. This phenomenon is most completely understood when the arguments are mutually independent random variables, and there exist several powerful complementary methods for proving concentration inequalities, such as the martingale method, the entropy method, and the method of transportation inequalities. The setting of dependent arguments is much less well understood. This chapter focuses on the martingale method for deriving concentration inequalities without independence assumptions. In particular, we use the machinery of so-called Wasserstein matrices to show that the Azuma-Hoeffding concentration inequality for martingales with almost surely bounded differences, when applied in a sufficiently abstract setting, is powerful enough to recover and sharpen several known concentration results for nonproduct measures. Wasserstein matrices provide a natural formalism for capturing the interplay between the metric and the probabilistic structures, which is fundamental to the concentration phenomenon.