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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 $$X$$.

##### MSC:
 60F05 Central limit and other weak theorems 60G60 Random fields
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