On weak convergence of approximate models of Gaussian random fields. (Russian) Zbl 0725.60045
Theory and applications of statistical modelling, Collect. Sci. Works, Novosibirsk, 31-39 (1988).
[For the entire collection see Zbl 0685.00022.]
Let u(x), \(x\in R^ n\), be a real Gaussian homogeneous random field with E u(x)\(=0\) and correlation function \(R(z)=\int_{R^ k}\cos <\lambda,z>\mu (d\lambda),\) \(z\in R^ k\), \[ u_ n(x)=\sum^{n}_{j=1}\sigma_ j(\xi_ j \cos <\lambda_ j,x>+\eta_ j \sin <\lambda_ j,x>) \] be an approximation for u(x), where \(\xi_ j,\eta_ j\) are independent Gaussian variables, and \[ R_ n(z)=cov(u_ n(0),u_ n(z))=\sum^{n}_{j=1}\sigma^ 2_ j \cos <\lambda_ j,z>. \] If for some \(\epsilon >0\), \(\int_{R^ k}| \lambda |^{\epsilon}\mu (d\lambda)<\infty\), then there exist a sequence of random fields \(u_ n(x)\) with a finite spectrum such that \(u_ n(x)\) weakly converges to u(x) as \(n\to \infty\). The case when \(\sigma^ 2_ j\) are random variables associated with measure \(\mu\) (\(\cdot)\) are considered, too.
Let u(x), \(x\in R^ n\), be a real Gaussian homogeneous random field with E u(x)\(=0\) and correlation function \(R(z)=\int_{R^ k}\cos <\lambda,z>\mu (d\lambda),\) \(z\in R^ k\), \[ u_ n(x)=\sum^{n}_{j=1}\sigma_ j(\xi_ j \cos <\lambda_ j,x>+\eta_ j \sin <\lambda_ j,x>) \] be an approximation for u(x), where \(\xi_ j,\eta_ j\) are independent Gaussian variables, and \[ R_ n(z)=cov(u_ n(0),u_ n(z))=\sum^{n}_{j=1}\sigma^ 2_ j \cos <\lambda_ j,z>. \] If for some \(\epsilon >0\), \(\int_{R^ k}| \lambda |^{\epsilon}\mu (d\lambda)<\infty\), then there exist a sequence of random fields \(u_ n(x)\) with a finite spectrum such that \(u_ n(x)\) weakly converges to u(x) as \(n\to \infty\). The case when \(\sigma^ 2_ j\) are random variables associated with measure \(\mu\) (\(\cdot)\) are considered, too.
Reviewer: N.Leonenko (Kiev)
