Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1539–1589 | Cite as

Asymptotic Behavior of Weighted Power Variations of Fractional Brownian Motion in Brownian Time

  • Raghid Zeineddine


We study the asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time \(Z_t:= X_{Y_t},t \geqslant 0\), where X is a fractional Brownian motion and Y is an independent Brownian motion.


Weighted power variations Limit theorem Malliavin calculus Fractional Brownian motion Fractional Brownian motion in Brownian time 

Mathematics Subject Classification 2010

60F05 60G15 60G22 60H05 60H07 



We are thankful to the referees for their careful reading of the original manuscript and for a number of suggestions. The financial support of the DFG (German Science Foundations) Research Training Group 2131 is gratefully acknowledged.


  1. 1.
    Gradinaru, M., Russo, F., Vallois, P.: Generalized covariations, local time and Stratonovich Itô’s formula for fractional Brownian motion with Hurst index \(H \ge 1/4\). Ann. Probab. 31(4), 1772–1820 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Khoshnevisan, D., Lewis, T.M.: Stochastic calculus for Brownian motion on a Brownian fracture. Ann. Appl. Probab. 9(3), 629–667 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Khoshnevisan, D., Lewis, T.M.: Iterated Brownian Motion and its Intrinsic skeletal structure. In: Dalang R.C., Dozzi M., Russo F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, vol 45. Birkhäuser, Basel (1999)Google Scholar
  4. 4.
    Nourdin, I., Peccati, G.: Weighted power variations of iterated Brownian motion. Electron. J. Probab. 13(43), 1229–1256 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Nourdin, I., Nualart, D., Tudor, C.: Central and non-central limit theorems for weighted power variations of the fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 46(4), 1055–1079 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Nourdin, I., Peccati, G.: Normal Approximations Using Malliavin Calculus: From Stein’s Method to the Universality. Cambridge University Press, Cambridge (2012)CrossRefMATHGoogle Scholar
  7. 7.
    Nourdin, I., Réveillac, A., Swanson, J.: The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6. Electron. J. Probab. 15(70), 2117–2162 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Nourdin, I., Zeineddine, R.: An Itô-type formula for the fractional Brownian motion in Brownian time. Electron. J. Probab. 19(99), 1–15 (2014)MATHGoogle Scholar
  9. 9.
    Zeineddine, R.: Fluctuations of the power variation of fractional Brownian motion in Brownian time. Bernoulli 21(2), 760–780 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Zeineddine, R.: Change-of-variable formula for the bi-dimensional fractional Brownian motion in Brownian time. ALEA Lat. Am. J. Probab. Math. Stat. 12(2), 597–683 (2015)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Research Training Group 2131, Fakultät MathematikTechnische Universität DortmundDortmundGermany

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