Uniform asymptotic normality of weighted sums of short-memory linear processes

TL;DR

This paper proves the uniform asymptotic normality of weighted sums of short-memory linear processes, with applications to regression and change point models, extending the understanding of their probabilistic behavior.

Contribution

It establishes the uniform asymptotic normality of weighted sums of short-memory linear processes for a broad class of functions, including applications to regression and change point analysis.

Findings

Convergence in outer distribution of weighted sums in Banach space

Application to regression models demonstrating asymptotic properties

Application to multiple change point models for statistical inference

Abstract

Let $X_1, X_2,\dots$ be a short-memory linear process of random variables. For $1\leq q<2$, let $\cF$ be a bounded set of real-valued functions on $[0,1]$ with finite $q$-variation. It is proved that $\{n^{-1/2}\sum_{i=1}^nX_if(i/n)\colon\,f\in\cF\}$ converges in outer distribution in the Banach space of bounded functions on $\cF$ as $n\to\infty$. Several applications to a regression model and a multiple change point model are given.