Stationary processes on the dyadic tree: prediction and covariance extension
The authors are interested in the study of linear systems indexed by the nodes of an infinite homogeneous tree of order 2. In their previous two papers [Integral Equations Oper.\ Theory 53, No.~1, 1--22 (2005; Zbl 1100.93010); C.~R., Math., Acad.\ Sci.\ Paris 336, No.~4, 293--298 (2003; Zbl 1030.46023)], they studied the stationary and nonstationary case and defined a corresponding function theory in each of these cases. In this third paper of the study, the case of stationary and isotropic stochastic processes of order \(q=2\) is considered. In this way, some necessary definitions such as point from the infinity, order on the tree, stationarity and prediction, and analogues of the Yule-Walker equations are introduced. The analogue of the band extension when the integers are replaced by the nodes of the dyadic tree is studied. A theorem for the existence and unicity of the band extension as a positive and stationary matrix is proved. In the last section, the special case of an isotropic process is analysed, using the obtained band extension methods.
scientific article; zbMATH DE number 5013374