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Functional central limit theorem for the measures of level surfaces of the Gaussian random field. (English. Russian original) Zbl 1278.60052
Theory Probab. Appl. 57, No. 1, 162-172 (2013); translation from Teor. Veroyatn. Primen. 57, No. 1, 168-178 (2012).
Let \(X=\{X_s, s\in \mathbb{R}^d\}\) be a centered stationary and isotropic Gaussian random field having \(C^1\) realizations. For \(t>0\) and \(x\in\mathbb{R}\) introduce the stochastic process \(N_t(x)=t^{-d/2}(\mathcal{H}_{d-1}(B_t(x)) - {\text{ E}}\mathcal{H}_{d-1}(B_t(x)))\) where \(B_t(x)=\{s\in [0,t]^d: X_s=x\}\) and \(\mathcal{H}_{d-1}\) is the \((d-1)\)-dimensional Hausdorff measure of a set in \(\mathbb{R}^d\). The authors prove that if the covariance function \(R\) of a field \(X\) satisfies certain conditions (involving first and second derivatives of \(R\)) then the family of random elements \(\{N_t, t>0\}\) converges in distribution as \(t\to \infty\) in the space \(L^2(\mathbb{R},\mu)\) to the specified centered Gaussian random element. Here \(\mu\) denotes the standard Gaussian measure on \(\mathbb{R}\). Thus the Hilbert space functional central limit theorem is established for the Hausdorff measures of the level sets of a field \(X\).

60F05 Central limit and other weak theorems
60G60 Random fields
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