On the convergence of the distributions of functionals of nonlinear transformations of Gaussian random fields. (Russian) Zbl 0577.60048

Leonenko, N. N.

Teor. Veroyatn. Mat. Stat. 29, 64-72 (1983).

Let \(\xi:\Omega \times {\mathbb{R}}^ n\to R^ 1\) be a homogeneous, isotropic and in the quadratic mean continuous Gaussian random field and r its not necessarily integrable correlation function. The random process \[ X_{\rho}(t):=\rho^{-n}r(\rho)^ k\int_{C(t,\rho)}H(\xi (x))dx \] is considered where \(t\in [0,1]\), C is a certain set, H a function from \(R^ 1\) to \(R^ 1\) with \(E(H(\xi (0))^ 2)<\infty\), and k is a rational integer connected with H. It is shown that \(X_{\rho}\) for \(\rho\) \(\to \infty\) weakly converges to a process, which is, depending on H, not always Gaussian and is given by multiple stochastic Itô-Wiener integrals.

Reviewer: C.Baldauf

MSC:

60G60	Random fields
60G15	Gaussian processes
60H05	Stochastic integrals

Keywords:

Gaussian random field; multiple stochastic Itô-Wiener integrals