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

Olenko, A. Ya.; Klykavka, B. M.

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 \(\mathbb R^n\).

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

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

Keywords:

random field; limit theorem; integral weight functional; spectral function; long range dependence

Citations:

Zbl 0539.60048