Stationary Processes Indexed by a Homogeneous Tree

Abstract

Let $T$ be the set of vertices of a homogeneous tree and let $(X_t)_{t\in T}$ be a second-order real or complex-valued process such that the expected value $\mathbb{E}(X_s\bar{X}_t)$ depends only on the distance between the vertices $s$ and $t$. In this paper we construct a measure space $(K, \mathscr{H}, m)$ and an isometry of the closed subspace of $L^2_\mathbb{C}(\Omega, \mathscr{A}, P)$ spanned by $(X_t)_{t\in T}$ onto $L^2(m)$.