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

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