Power variation for Gaussian processes with stationary increments

Abstract

We develop the asymptotic theory for the realised power variation of the processes X = ϕ • G , where G is a Gaussian process with stationary increments. More specifically, under some mild assumptions on the variance function of the increments of G and certain regularity conditions on the path of the process ϕ we prove the convergence in probability for the properly normalised realised power variation. Moreover, under a further assumption on the Hölder index of the path of ϕ , we show an associated stable central limit theorem. The main tool is a general central limit theorem, due essentially to Hu and Nualart [Y. Hu, D. Nualart, Renormalized self-intersection local time for fractional Brownian motion, Ann. Probab. (33) (2005) 948–983], Nualart and Peccati [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. (33) (2005) 177–193] and Peccati and Tudor [G. Peccati, C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, in: M. Emery, M. Ledoux, M. Yor (Eds.), Seminaire de Probabilites XXXVIII, in: Lecture Notes in Math, vol. 1857, Springer-Verlag, Berlin, 2005, pp. 247–262], for sequences of random variables which admit a chaos representation.