On Strong Embeddings by Stein's Method

TL;DR

This paper extends Stein's method to a broader class of discrete distributions, providing logarithmic rates for strong embeddings of partial sums to Brownian motion and bridges, with applications in probability and statistics.

Contribution

It generalizes Chatterjee's approach from binary variables to general discrete distributions and establishes new coupling rates for partial sums and exchangeable variables.

Findings

Logarithmic coupling rates for partial sums to Brownian motion.

Coupling results for exchangeable variables to Brownian bridges.

Extension of Stein's method to general discrete distributions.

Abstract

Strong embeddings, that is, couplings between a partial sum process of a sequence of random variables and a Brownian motion, have found numerous applications in probability and statistics. We extend Chatterjee's novel use of Stein's method for $\{-1,+1\}$ valued variables to a general class of discrete distributions, and provide $\log n$ rates for the coupling of partial sums of independent variables to a Brownian motion, and results for coupling sums of suitably standardized exchangeable variables to a Brownian bridge.