Hyper-positive definite functions I: scalar case, branching-type stationary stochastic processes

TL;DR

This paper introduces new concepts of branching-type stationary stochastic processes and hyper-positivity, providing criteria for their existence and properties on rooted trees, with applications to hypercontractive inequalities for Hankel operators.

Contribution

It defines branching-type processes and hyper-positivity, establishes existence criteria, and connects these to classical theorems and inequalities in operator theory.

Findings

Necessary and sufficient condition for processes on rooted trees

Complete characterization of hyper-positive functions on homogeneous trees

Hypercontractive inequalities for Hankel operators with hyper-positive symbols

Abstract

We propose a definition of branching-type stationary stochastic processes on rooted trees and related definitions of hyper-positivity for functions on the unit circle and functions on the set of non-negative integers. We then obtain (1) a necessary and sufficient condition on a rooted tree for the existence of non-trivial branching-type stationary stochastic processes on it, (2) a complete criterion of the hyper-positive functions in the setting of rooted homogeneous trees in terms of a variant of the classical Herglotz-Bochner Theorem, (3) a prediction theory result for branching-type stationary stochastic processes. As an unexpected application, we obtain natural hypercontractive inequalities for Hankel operators with hyper-positive symbols.