Probability Theory and Related Fields

, Volume 142, Issue 3, pp 339–366

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

Article

DOI: 10.1007/s00440-007-0107-9

Cite this article as:
Csörgő, M. & Kulik, R. Probab. Theory Relat. Fields (2008) 142: 339. doi:10.1007/s00440-007-0107-9

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.

Keywords

Long range dependenceLinear processesBahadur–Kiefer processQuantile processesStrong approximation

Mathematics Subject Classification (2000)

60F1560F17

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia
  3. 3.Mathematical InstituteWrocław UniversityWrocławPoland