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~\emph{randomized pivot order}, thereby answering an open question raised in the same article. We deduce this under minimal and global assumptions on involving {\em 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 {\em smaller} than the previously known upper bound. The main ingredients are a simplified \emph{skeletal} process approximating the GSW design and concentration phenomena for random matrices obtained from random sampling using the Stein's method for exchangeable pairs

翻译：本文证明了基于Gram–Schmidt游走（GSW）设计的Horvitz–Thompson估计量的中心极限定理，该设计由Harshaw等人（2022）最新提出。我们特别考虑了采用随机化枢轴次序的GSW设计版本，从而回答了同一篇文章中提出的开放性问题。我们仅在涉及问题参数（如（总和）潜在结果向量和协变量矩阵）的最小全局假设下推导出该定理。作为分析的一个有趣推论，我们还获得了用这些参数表示的估计量的精确极限方差，该方差小于先前已知的上界。主要创新点包括一个简化骨架过程来近似GSW设计，以及利用斯坦因交换对方法从随机抽样得到的随机矩阵的集中现象。