Probability Theory and Related Fields

, Volume 142, Issue 3, pp 339-366

First online:

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

  • Miklós CsörgőAffiliated withSchool of Mathematics and Statistics, Carleton University
  • , Rafał KulikAffiliated withSchool of Mathematics and Statistics, University of SydneyMathematical Institute, Wrocław University Email author 

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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.


Long range dependence Linear processes Bahadur–Kiefer process Quantile processes Strong approximation

Mathematics Subject Classification (2000)

60F15 60F17