The functional convergence of the limits and models in the Monte Carlo method (Q1311536)

Let \(\xi(x)\) be a real Gaussian uniform random field with spectral density \(p(\lambda)\); \(E\psi(x)= 0\), \(\text{var }\psi(x)= 1\), \(x\in X\), \(X\) being a compact in \(\mathbb{R}^ k\). Let us consider the partition of the spectral space \(\Lambda\in \mathbb{R}^ k\) into non-intersecting subregions \(\Lambda_ 1,\dots,\Lambda_ n\), and as an approximation model of the field \(\xi(x)\) consider the expression \[ \xi(x)= \sum^ n_{j=1} \sqrt{p_ j} \bigl(\nu_ j\sin\langle \lambda_ j,x\rangle+ \eta_ j\cos \langle\lambda_ j,x\rangle\bigr),\tag{1} \] where \(\lambda_ j\in \Lambda_ j\), \(\nu_ j\), \(\eta_ j\) are iid random variables with zero mean and unit variances, \(p_ j=\int_{\Lambda_ j} p(\lambda)d\lambda\) and \(\langle.,.\rangle\) denotes the scalar product in \(\mathbb{R}^ k\). It is assumed that \(\nu_ j\) and \(\eta_ j\) are Gaussian. The following assertion is proved. If \(\int_ \Lambda \|\lambda\|^ \beta p(\lambda)d\lambda< \infty\) for some \(\beta>0\), the randomized spectral model (1) which satisfies \[ \max_{j=1,\dots,n-1}\text{diam}\bigl\{\lambda: \|\lambda\|\geq n\bigr\}\to\infty\qquad\text{as}\quad c_ n\to \infty, \] converges weakly in \(C_ x\) to the uniform Gaussian field \(\xi\) with spectral density \(p(\lambda)\). In the sequel the method is an approach to justify the method of dependent tests for a multidimensional parameter and to investigate the convergence of local limits of the solution of an integral equation of the second kind.