One reason why standard formulations of the central limit theorems are not applicable in high-dimensional and non-stationary regimes is the lack of a suitable limit object. Instead, suitable distributional approximations can be used, where the approximating object is not constant, but a sequence as well. We extend Gaussian approximation results for the partial sum process by allowing each summand to be multiplied by a data-dependent matrix. The results allow for serial dependence of the data, and for high-dimensionality of both the data and the multipliers. In the finite-dimensional and locally-stationary setting, we obtain a functional central limit theorem as a direct consequence. An application to sequential testing in non-stationary environments is described.

翻译：标准中心极限定理的表述在高维和非平稳情形下不适用，其原因之一在于缺乏合适的极限对象。此时可采用分布逼近方法，其中逼近对象并非常数，而同样是一个序列。我们通过允许每个求和项乘以一个数据依赖的矩阵，推广了部分和过程的高斯逼近结果。该结果允许数据存在序列相关性，且支持数据与乘子的高维特性。在有限维和局部平稳设定下，我们直接得到了泛函中心极限定理。文中还描述了该方法在非平稳环境下序列检验中的应用。