Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition

TL;DR

This paper develops a multivariate exchangeable pairs approach within Stein's method to approximate potentially singular multivariate normal distributions, introducing an embedding technique for broader applicability.

Contribution

It extends Stein's method with exchangeable pairs to handle singular distributions and proposes an embedding method for complex statistics.

Findings

Effective normal approximation for singular multivariate distributions

Application to runs on the line and permutation statistics

Extension of Stein's method with a new embedding technique

Abstract

In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein's method to assess distributional distances to potentially singular multivariate normal distributions. By extending the statistics into a higher-dimensional space, we also propose an embedding method which allows for a normal approximation even when the corresponding statistics of interest do not lend themselves easily to Stein's exchangeable pairs approach. To illustrate the method, we provide the examples of runs on the line as well as double-indexed permutation statistics.