Abstract
We give a new proof and provide new bounds for the speed of convergence in the Central Limit Theorem of Breuer Major on stationary Gaussian time series, which generalizes to particular triangular arrays. Our assumptions are given in terms of the spectral density of the time series. We then consider generalized quadratic variations of Gaussian fields with stationary increments under the assumption that their spectral density is asymptotically self-similar and prove Central Limit Theorems in this context.
Highlights
In this paper we essentially develop Central Limit Theorems that are well adapted to obtain asymptotic properties of quadratic variations of Gaussian fields with stationary increments
From its spectral measure τ, which is such that vY (t) = 2 |e−it·x − 1|2 dτ(x), ∀t ∈ Rν
We want to have Central Limit Theorems for the quadratic variation of this sequence when n tends to ∞. This quadratic variation is related to the means of a discrete time series, whose spectral density is obtained by periodization of F
Summary
In this paper we essentially develop Central Limit Theorems that are well adapted to obtain asymptotic properties of quadratic variations of Gaussian fields with stationary increments. Anisotropic but still self-similar generalizations are obtained by considering a spectral density given by F (x) = Ω(x)FH (x) with Ω an homogeneous function of degree 0 satisfying Ω(x) = Ω(x/|x|). We want to have Central Limit Theorems for the quadratic variation of this sequence when n tends to ∞ For fixed n, this quadratic variation is related to the means of a discrete time series, whose spectral density is obtained by periodization of F. Once we have Central Limit Theorems for finite distributions through this scaling argument, we can recover asymptotic properties for continuous time quadratic variations, which may be used when dealing with increments of non linear functionals of Y instead of increments of Y.
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