Reduction principles for quantile and Bahadur-Kiefer processes of long-range dependent linear sequences

TL;DR

This paper studies weighted quantile and Bahadur-Kiefer processes for long-range dependent linear sequences across the entire interval, revealing unique asymptotic behaviors not seen in i.i.d. cases.

Contribution

It extends the analysis of quantile and Bahadur-Kiefer processes to long-range dependent sequences on the full interval with weights, uncovering new asymptotic phenomena.

Findings

Weak convergence of Bahadur-Kiefer processes

Distinct pointwise behaviors of processes

Unusual behavior of the general quantile process

Abstract

In this paper we consider quantile and Bahadur-Kiefer processes for long range dependent linear sequences. These processes, unlike in previous studies, are considered on the whole interval $(0,1)$. As it is well-known, quantile processes can have very erratic behavior on the tails. We overcome this problem by considering these processes with appropriate weight functions. In this way we conclude strong approximations that yield some remarkable phenomena that are not shared with i.i.d. sequences, including weak convergence of the Bahadur-Kiefer processes, a different pointwise behavior of the general and uniform Bahadur-Kiefer processes, and a somewhat "strange" behavior of the general quantile process.