Probability Theory and Related Fields

, Volume 159, Issue 1–2, pp 237–272 | Cite as

Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion

  • Krzysztof Burdzy
  • David Nualart
  • Jason SwansonEmail author


The purpose of this paper is to study the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter \(H=1/6\). We prove that, under some conditions on both subsequences, the limit is a two-dimensional Brownian motion whose components may be correlated and we find explicit formulae for its covariance function.


Fractional Brownian motion Cubic variation Convergence in law 

Mathematics Subject Classification

Primary 60G22; Secondary 60F17 



The authors are grateful for the careful reading and helpful suggestions of an anonymous referee. Krzysztof Burdzy was partially supported by Grant DMS-1206276 from the NSF and by Grant N N201 397137 from the MNiSW, Poland. David Nualart was partially supported by Grant DMS-1208625 from the NSF.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Krzysztof Burdzy
    • 1
  • David Nualart
    • 2
  • Jason Swanson
    • 3
    Email author
  1. 1.University of WashingtonSeattleUSA
  2. 2.University of KansasLawrenceUSA
  3. 3.University of Central FloridaOrlandoUSA

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