MULTISCALE AUTOREGRESSIVE PROCESSES .2. LATTICE STRUCTURES FOR WHITENING AND MODELING

In part I of this two-part paper we introduced a class of stochastic processes defined on dyadic homogenous trees. The motivation for the study of these processes comes from our desire to develop a theory for multiresolution descriptions of stochastic processes in one and multiple dimensions based on the idea underlying the recently introduced theory of wavelet transforms. In part I we described how this objective leads to the study of processes on trees and began the development of a theory of autoregressive (AR) models for isotropic processes on trees. In this second part we complete that investigation by developing lattice structures for the whitening and modeling of isotropic processes on trees. We also present a result relating the stability properties of these models to the reflection coefficient sequence introduced in part I. In addition, this framework allows us to obtain a detailed analysis of the Wold decomposition of processes on trees. One interesting aspect of this is that there is a significantly larger class of singular processes on dyadic trees than on the integers.