Central limit theorem for Gram-Schmidt random walk design
Sabyasachi Chatterjee, Partha S. Dey, Subhajit Goswami · The Annals of Applied Probability · 2025
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. (J. Amer. Statist. Assoc. 119 (2024) 2934–2946). In particular, we consider the version of GSW design, which uses a randomized pivot order. We deduce our result under very mild assumptions involving only the problem parameters, such as the (sum) potential outcome vector and the covariate matrix. As a very important consequence of our analysis, we 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 Stein’s method for exchangeable pairs.