Second order random fields over l p with homogeneous and isotropic increments

Abstract

A random field over l p is a stochastic process X(t), where t is an element of l p .It is said to have homogeneous and isotropic increments if E(X(t) − X(s)) 2 is a function of ∥t-s∥. The subject of this work is the spectral theory of such processes. The main results are: a representation of the field as a series of filtered, orthogonal processes with a real time parameter; a representation as a white noise integral over l p ;limit theorems for the average of X over a sphere; and, finally, filtering of the orthogonal components. In particular, we mention: (1) The averages over spheres of increasing dimension converge in quadratic mean for p=2 but not for 0<p<2. (2) The limiting distribution of a fixed coordinate of a point uniformly distributed over the l p -unit sphere in n-space, n → ∞, has the density [2γ(1/p+1)p 1/p ]−1exp(-¦x¦ p /P).