# DIMENSION WALKS AND SCHOENBERG SPECTRAL MEASURES FOR ISOTROPIC RANDOM FIELDS

*by D. J. Daley*

*Institution:*The University of Melbourne

*Date: Thu 10th May 2012*

*Time: 4:15 PM*

*Location: Old Geology-Theatre 2*

*Abstract*: Schoenberg (1938) showed how Bochner's basic representation theorem for

positive definite functions (e.g. correlation function of a stationary

stochastic process) `simplifies' for spatial processes (d-dimensional random

fields) which are isotropic: the standard Fourier kernel function is replaced

by the characteristic function of a random direction in d-space and the

spectral measure, instead of being on d-space, is on the positive half-line:

we call this a d-Schoenberg measure.

The talk describes how Wendland's `dimension walks', which were defined

earlier by Matheron as Descente and Montee in studying relations between d-D

and either (d+2)-D or (d-2)-D correlation functions, are equivalent to

simple modifications of their d-Schoenberg measures.

Another family of dimension walks arises from projections from unit d-spheres

to lower dimensional spheres, first via the kernel functions in the

Schoenberg representation and then more generally, for d-Schoenberg measures.