## A limit theorem for random fields with a singularity in the spectrum.(Ukrainian, English)Zbl 1224.60115

Teor. Jmovirn. Mat. Stat. 81, 128-138 (2009); translation in Theory Probab. Math. Stat. 81, 147-158 (2010).
Let $$\xi(x),\;x\in\mathbb R^n$$, be a real-valued measurable mean square continuous wide sense homogeneous and isotropic Gaussian random field [M. I. Yadrenko, Spectral theory of random fields. New York etc.: Springer-Verlag (1983; Zbl 0539.60048)] with the zero mean, covariance function $$B_n(r) = B_n(| x| ) = E\xi(0)\xi(x)$$, $$| x| =(\sum_{i=1}^nx_i^2)^{1/2}$$, and spectral density $$\varphi(\lambda)=(h(|\lambda| -a))/| | \lambda| -a| ^{1-\alpha}$$, where $$0<\alpha<1$$ and $$h$$ is a function defined and bounded in the interval $$[-a,+\infty)$$, continuous in a neighborhood of $$0$$ and such that $$h(0)\not=0$$. This class of fields generalizes the case of random fields with long range dependence, where the spectrum has a singularity at the origin. The authors prove that the finite dimensional distributions of the process $X_r(t) =\frac{\sqrt{t}r^{\alpha/2}}{(2\pi)^{n/2}\sqrt{2h(0)}a^{(n-1)/2}} \int_{\mathbb R^n}f_{n,rt^{1/n},a}(| x| )\xi(x)\,dx$ weakly converge as $$r\to\infty$$ to the finite-dimensional distributions of the process $X(t) =\int_{\mathbb R^n} \frac{J_{n/2}(| u| t^{1/n})} {| u| ^{n-\alpha/2}}dZ(u),$ where $f_{n,r,a}(| x| ) =\frac{1}{| x| ^{n/2-1}} \int_0^{\infty}\lambda^{n/2}\frac{J_{n/2}(r(\lambda-a))}{(r(\lambda-a))^{n/2}}J_{n/2-1}(| x| \lambda)d\lambda,\quad | x| \not=r,$ and where $$Z(u)$$ is the white noise in $$\mathbb R^n$$.

### MSC:

 60G60 Random fields 60F17 Functional limit theorems; invariance principles

Zbl 0539.60048
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