We study the universality property of estimators for high-dimensional linear models, which implies that the distribution of estimators is independent of whether the covariates follow a Gaussian distribution. Recent developments in high-dimensional statistics typically require covariates to strictly follow a Gaussian distribution to precisely characterize the properties of estimators. To relax this Gaussianity requirement, the existing literature has examined conditions under which estimators achieve universality. In particular, independence among the elements of the high-dimensional covariates has played a critical role. In this study, we focus on high-dimensional linear models with covariates exhibiting block dependence, where covariate elements can only be dependent within each block, and show that estimators for such models retain universality. Specifically, we prove that the distribution of estimators with Gaussian covariates can be approximated by the distribution of estimators with non-Gaussian covariates having the same moments under block dependence. To establish this result, we develop a generalized Lindeberg principle suitable for handling block dependencies and derive new error bounds for correlated covariate elements. We further demonstrate the universality result across several different estimators.

翻译：我们研究了高维线性模型估计量的普适性性质，该性质意味着估计量的分布与协变量是否服从高斯分布无关。高维统计学的最新进展通常要求协变量严格服从高斯分布，以精确刻画估计量的性质。为了放宽这一高斯性要求，现有文献已探讨了估计量达到普适性所需的条件。特别地，高维协变量各元素之间的独立性在其中起到了关键作用。在本研究中，我们关注协变量呈现块依赖性的高维线性模型，其中协变量元素仅能在每个块内具有依赖性，并证明此类模型的估计量仍保持普适性。具体而言，我们证明了具有高斯协变量的估计量的分布，可以用具有相同矩且在块依赖结构下的非高斯协变量的估计量分布来近似。为建立这一结果，我们发展了一种适用于处理块依赖性的广义林德伯格原理，并为相关的协变量元素推导了新的误差界。我们进一步在几种不同的估计量上验证了这一普适性结果。