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Isotropic correlation functions on \(d\)-dimensional balls. (English) Zbl 0944.60025

Summary: A popular procedure in spatial data analysis is to fit a line segment of the form \(c(x)= 1-\alpha\|x\|\), \(\|x\|< 1\), to observed correlations at (appropriately scaled) spatial lag \(x\) in \(d\)-dimensional space. We show that such an approach is permissible if and only if \[ 0\leq \alpha\leq {2\Gamma(d/2)\over \pi^{1/2}\Gamma((d+ 1)/2)}, \] the upper bound depending on the spatial dimension \(d\). The proof relies on G. Matheron’s turning bands operator [Adv. Appl. Probab. 5, 439-468 (1973; Zbl 0324.60036)] and an extension theorem for positive definite functions due to W. Rudin [Duke Math. J. 37, 49-53 (1970; Zbl 0194.36002)]. Side results and examples include a general discussion of isotropic correlation functions defined on \(d\)-dimensional balls.

MSC:

60D05 Geometric probability and stochastic geometry
86A32 Geostatistics
60G60 Random fields
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