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Quadratic variations and estimation of the local Hölder index of a Gaussian process. (English) Zbl 0882.60032

Let \(X\) be a Gaussian stochastic process with stationary increments which is observed at times \(j\Delta, j=1,\dots,n\). For a finite sequence of real numbers \(a_0,\dots,a_p\) with zero sum the quadratic \(a\)-variation is defined by \[ V(a,n,\Delta)=(1/n)\sum_{j=1}^{n-p}\left(\left(\Delta_a X_j^2 /\sigma^2_{a,\Delta} \right) -1\right),\quad \Delta_a X_j =\sum_{i=0}^p a_iX((i+j)\Delta), \] where \(\sigma^2_{a,\Delta}\) denotes the variance of \(\Delta_a X_j\). Conditions are given that ensure the almost sure convergence as \(n\to \infty\) of the quadratic \(a\)-variations. Some stronger conditions are given that ensure the asymptotic normality of the quadratic \(a\)-variations. A method is proposed for the identification of the local Hölder index of a Gaussian process from a discrete observation of one sample path. A convergence theorem and a central limit theorem for the estimator of this index are obtained.

MSC:

60G12 General second-order stochastic processes
60G35 Signal detection and filtering (aspects of stochastic processes)
60F05 Central limit and other weak theorems
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