Non-central limit theorem for the cubic variation of a class of selfsimilar stochastic processes

Khalifa Es-Sebaiy (1) , Ciprian A. Tudor (2)

1 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
2 LPP - Laboratoire Paul Painlevé - UMR 8524

By using multiple Wiener-Itô stochastic integrals, we study the cubic variation of a class of selfsimilar stochastic processes with stationary increments (the Rosenblatt process with selfsimilarity order $H\in (\frac{1}{2}, 1)$). This study is motivated by statistical purposes. We prove that this renormalized cubic variation satisfies a non-central limit theorem and its limit is (in the $L^{2}(\Omega)$ sense) still the Rosenblatt process.

Domaines: Probabilités [math.PR]

Khalifa Es-Sebaiy, Ciprian A. Tudor. Non-central limit theorem for the cubic variation of a class of selfsimilar stochastic processes. SIAM Theory of Probability and its Applications, 2010, 55 (3), pp.1-23. ⟨hal-00432690⟩