Radial Positive Definite Functions Generated by Euclid's Hat

✍ Scribed by Tilmann Gneiting

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 305 KB
• Volume: 69
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1998.1800

✦ Synopsis

Radial positive definite functions are of importance both as the characteristic functions of spherically symmetric probability distributions, and as the correlation functions of isotropic random fields. The Euclid's hat function h n (&x&), x # R n , is the self-convolution of an indicator function supported on the unit ball in R n . This function is evidently radial and positive definite, and so are its scale mixtures that form the class H n . Our main results characterize the classes H n , n 1, and H = n 1 H n . This leads to an analogue of Po lya's criterion for radial func-

is convex for k=[(n&2)Â2], the greatest integer less than or equal to (n&2)Â2, then .(&x&) is a characteristic function in R n . Along the way, side results on multiply monotone and completely monotone functions occur. We discuss the relations of H n to classes of radial positive definite functions studied by Askey (Technical