Strong Gaussian approximations with random multipliers

TL;DR

This paper extends Gaussian approximation results to high-dimensional, non-stationary data by incorporating data-dependent multipliers, enabling better distributional approximations and applications like sequential testing.

Contribution

It introduces a novel Gaussian approximation framework that accounts for data-dependent multipliers in high-dimensional, dependent, and non-stationary settings.

Findings

Allows for serial dependence in data

Handles high-dimensional multipliers

Provides a functional central limit theorem

Abstract

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.