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Reduction principles for quantile and Bahadur–Kiefer processes of long-range dependent linear sequences
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  • Published: 30 October 2007

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

  • Miklós Csörgő1 &
  • Rafał Kulik2,3 

Probability Theory and Related Fields volume 142, pages 339–366 (2008)Cite this article

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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.

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Miklós Csörgő

  2. School of Mathematics and Statistics, University of Sydney, NSW, 2006, Sydney, Australia

    Rafał Kulik

  3. Mathematical Institute, Wrocław University, Pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Rafał Kulik

Authors
  1. Miklós Csörgő
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  2. Rafał Kulik
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Corresponding author

Correspondence to Rafał Kulik.

Additional information

Research supported in part by NSERC Canada Discovery Grants of Miklós Csörgő, Donald Dawson and Barbara Szyszkowicz at Carleton University.

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Csörgő, M., Kulik, R. Reduction principles for quantile and Bahadur–Kiefer processes of long-range dependent linear sequences. Probab. Theory Relat. Fields 142, 339–366 (2008). https://doi.org/10.1007/s00440-007-0107-9

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  • Received: 02 April 2007

  • Revised: 26 September 2007

  • Published: 30 October 2007

  • Issue Date: November 2008

  • DOI: https://doi.org/10.1007/s00440-007-0107-9

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Keywords

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

Mathematics Subject Classification (2000)

  • 60F15
  • 60F17
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