Limit Theorems for Quadratic Variations of the Lei–Nualart Process

  • Salwa Bajja
  • Khalifa Es-Sebaiy
  • Lauri Viitasaari
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 271)


Let X be a Lei–Nualart process with Hurst index \(H\in (0, 1)\), \(Z_{1}\) be an Hermite random variable. For any \(n \ge 1\), set
$$V_{n}=\sum _{k=0}^{n-1}\left[ n^{2H}(\varDelta _k X)^{2}-n^{2H}\mathrm {I\! E}(\varDelta _k X)^{2}\right] .$$
The aim of the current paper is to derive, in the case when the Hurst index verifies \(H > 3/4\), an upper bound for the total variation distance between the laws \(\mathcal {L}(Z_n)\) and \(\mathcal {L}(Z_{1})\), where \(Z_{n}\) stands for the correct renormalization of \(V_{n}\) which converges in distribution towards \(Z_{1}\). We derive also the asymptotic behavior of quadratic variations of process X in the critical case \(H=3/4\), i.e. an upper bound for the total variation distance between the \(\mathcal {L}(Z_n)\) and the Normal law.


Hermite random variable Gaussian analysis Malliavin calculus Convergence in law Berry–Esseen bounds 


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Salwa Bajja
    • 1
  • Khalifa Es-Sebaiy
    • 2
  • Lauri Viitasaari
    • 3
  1. 1.National School of Applied Sciences—MarrakeshCadi Ayyad UniversityMarrakeshMorocco
  2. 2.Department of Mathematics, Faculty of ScienceKuwait UniversityKuwaitKuwait
  3. 3.Department of Mathematics and System AnalysisAalto University School of ScienceAaltoFinland

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