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.

翻译：我们证明了基于Harshaw等人(2022)近期提出的格莱姆-施密特随机游走(GSW)设计的霍维茨-汤普森估计量的中心极限定理。特别地，本文考虑采用随机枢轴顺序的GSW设计版本，从而回答了该文献中提出的一个开放性问题。我们在仅涉及问题参数（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。作为我们分析的一个有趣推论，我们还获得了这些参数下估计量的精确极限方差，该方差小于先前已知的上界。主要技术手段包括：近似GSW设计的简化骨架过程，以及采用施泰因交换对方法从随机抽样获得的随机矩阵的集中现象。