Central Limit Theorem for Gram-Schmidt Random Walk Design

TL;DR

This paper establishes a central limit theorem for the Horvitz-Thompson estimator under the Gram-Schmidt Walk design with randomized pivot order, providing new insights into its variance and asymptotic behavior.

Contribution

It proves a CLT for the GSW design with randomized pivots under minimal assumptions, answering an open question and deriving the exact limiting variance.

Findings

CLT for the GSW design with randomized pivots

Explicit expression for the limiting variance

Variance is smaller than previous upper bounds

Abstract

We prove a central limit theorem for the Horvitz-Thompson estimator based on the Gram-Schmidt Walk (GSW) design, recently developed in Harshaw et al.(2022). In particular, we consider the version of the GSW design which uses randomized pivot order, thereby answering an open question raised in the same article. We deduce this under minimal and global assumptions involving only the problem parameters such as the (sum) potential outcome vector and the covariate matrix. As an interesting consequence of our analysis we also obtain the precise limiting variance of the estimator in terms of these parameters which is smaller than the previously known upper bound. The main ingredients are a simplified skeletal process approximating the GSW design and concentration phenomena for random matrices obtained from random sampling using the Stein's method for exchangeable pairs.