Invariance principles for linear processes with application to isotonic regression

TL;DR

This paper establishes maximal inequalities and a functional central limit theorem for linear processes with dependent innovations, applying these results to analyze isotonic regression estimators under long-range dependence.

Contribution

It introduces new maximal inequalities and a CLT for linear processes with general weights, extending the analysis to long-range dependent error processes in isotonic regression.

Findings

Limit distribution is fractional Brownian motion.

Provides new approximation techniques for dependent linear processes.

Analyzes isotonic regression with long-range dependent errors.

Abstract

In this paper, we prove maximal inequalities and study the functional central limit theorem for the partial sums of linear processes generated by dependent innovations. Due to the general weights, these processes can exhibit long-range dependence and the limiting distribution is a fractional Brownian motion. The proofs are based on new approximations by a linear process with martingale difference innovations. The results are then applied to study an estimator of the isotonic regression when the error process is a (possibly long-range dependent) time series.