# Department Seminars and Colloquia

### DIMENSION WALKS AND SCHOENBERG SPECTRAL MEASURES FOR ISOTROPIC RANDOM FIELDS

Stochastic Processes Seminar

#### 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.