On some properties of probability metrics (Q2740087)

The authors consider the problem of closeness in different metrics of spectral and correlation functions of random fields. Let \(\gamma_1(x), \gamma_2(x), x\in R^{n}\), be real, measurable, mean-square continuous, homogeneous isotropic random fields. Denote by \(B_{n,1}(t), B_{n,2}(t)\) their correlation functions and by \(\Phi_{n,1}(\lambda), \Phi_{n,2}(\lambda)\) the corresponding spectral functions. One of the presented results is the following: If \(k_1(\Phi_{n,1},\Phi_{n,2})<\infty\) and there exists \(H>0\) such that \(\forall r\in [0,H]\;B_{n,1}(r)= B_{n,2}(r)\), then for all \(y>0\), NEWLINE\[NEWLINE\chi_1(B_{n,1}, B_{n,2})\leq {\Gamma(n/2)\over \Gamma((n+1)/2)\sqrt{\pi}}T_{n}(H,y,\Phi_{n,1},\Phi_{n,2}),NEWLINE\]NEWLINE where NEWLINE\[NEWLINE T_{n}(H,y,F_{1},F_{2})=\min\left\{{{\pi}\over{H}};\int_{y}^{+\infty} |F_{1}(u)-F_{2}(u)|du+ 3(y+2)\sqrt{2}\left({\pi\sqrt{n}\over H}\right)^{n/(n+1)}\right\};NEWLINE\]NEWLINE \(k_1(F_{\xi},F_{\eta})=\int_{R}|F_{\xi}-F_{\eta}|dx\) is the average metric for one-dimensional distribution functions; \(\chi_{s}(\varphi_{\xi}, \varphi_{\eta})=\sup_{t\in R}|\varphi_{\xi}- \varphi_{\eta}|/|t|^{s}\) is the weighted uniform metric for characteristic functions.