Lithuanian Mathematical Journal, Vol. 54, No. 3, July, 2014, pp. 323–344 A central limit theorem for a weighted power variation of a Gaussian process ∗ Raimondas Malukas and Rimas Norvaiša Institute of Mathematics and Informatics, Vilnius University, Akademijos str. 4, LT-08663 Vilnius, Lithuania (e-mail: raimma@gmail.com; rimas.norvaisa@mii.vu.lt) Received November 28, 2013; revised March 12, 2014 Abstract. Let ρ be a real-valued function on [0,T ], and let LSI (ρ) be a class of Gaussian processes over time interval [0,T ], which need not have stationary increments but their incremental variance σ(s, t) is close to the values ρ(|t − s|) as t → s uniformly in s ∈ (0,T ]. For a Gaussian processes G from LSI (ρ), we consider a power variation V n corresponding to a regular partition π n of [0,T ] and weighted by values of ρ(·). Under suitable hypotheses on G, we prove that a central limit theorem holds for V n as the mesh of π n approaches zero. The proof is based on a general central limit theorem for random variables that admit a Wiener chaos representation. The present result extends the central limit theorem for a power variation of a class of Gaussian processes with stationary increments and for bifractional and subfractional Gaussian processes. MSC: 60G15, 60G22, 60F05 Keywords: weighted power variation, central limit theorem, Gaussian processes, locally stationary increments, bifrac- tional Gaussian process, subfractional Gaussian process 1 Introduction Let ρ be a real-valued function on [0,T ], T> 0, and let G = {G(t): t ∈ [0,T ]} be a zero-mean Gaussian stochastic process such that its incremental variance σ G (s, t)=(E[G(t) − G(s)] 2 ) 1/2 , s, t ∈ [0,T ], is close to the values ρ(|t − s|) as t → s uniformly in s ∈ (0,T ]. The exact meaning of this assumption is formulated by Definition 1 below describing the class LSI (ρ) of Gaussian processes with local variance ρ. Let (m n ) n∈N be an increasing and unbounded sequence of positive integers. For each n ∈ N, t n i := iT /m n , i =0, 1,...,m n , are equally spaced points of [0,T ] making its regular partition π n , and its mesh Δ n := T/m n = t n i − t n i−1 → 0 as n →∞. Given an r> 0, we are interested in the asymptotic behavior as n →∞ of the sums V n := V (G, r, ρ, m n ) := mn  i=1  |G(t n i ) − G(t n i−1 )| ρ(Δ n )  r Δ n , (1.1) called the weighted rth-power variation. The local variance ρ plays the role of the weight function in the preceding sum. * This research was funded by a grant (No. MIP-053/2012) from the Research Council of Lithuania. 323 0363-1672/14/5403-0323 c  2014 Springer Science+Business Media New York