Stein's method of exchangeable pairs in multivariate functional approximations

TL;DR

This paper extends Stein's method using exchangeable pairs for multivariate functional approximation, applying it to random graph subgraph counts and U-processes, including cases beyond current theoretical limits.

Contribution

It develops a new framework for multivariate functional Gaussian approximation using exchangeable pairs, expanding applicability to complex dependent structures.

Findings

Provides bounds for Gaussian approximation of subgraph counts in Erdős-Rényi graphs.

Derives bounds for vectors of weighted U-processes, including single-run cases.

Demonstrates the method's applicability to complex dependent data structures.

Abstract

In this paper we develop a framework for multivariate functional approximation by a suitable Gaussian process via an exchangeable pairs coupling that satisfies a suitable approximate linear regression property, thereby building on work by Barbour (1990) and Kasprzak (2020). We demonstrate the applicability of our results by applying it to joint subgraph counts in an Erd\H{o}s-Renyi random graph model on the one hand and to vectors of weighted, degenerate $U$-processes on the other hand. As a concrete instance of the latter class of examples, we provide a bound for the functional approximation of a vector of success runs of different lengths by a suitable Gaussian process which, even in the situation of just a single run, would be outside the scope of the existing theory.