Random Gaussian sums on trees

TL;DR

This paper studies Gaussian processes on trees, providing conditions for their almost sure boundedness based on tree structure and weights, with complete characterizations for binary trees and homogeneous weights.

Contribution

It offers necessary and sufficient conditions for Gaussian process boundedness on trees and fully characterizes weights ensuring boundedness in specific cases.

Findings

Conditions for a.s. boundedness of Gaussian processes on trees

Characterization of weights for boundedness on binary trees

Analysis of the metric properties related to process boundedness

Abstract

Let $T$ be a tree with induced partial order $\preceq$. We investigate centered Gaussian processes $X=(X_t)_{t\in T}$ represented as $$ X_t=\sigma(t)\sum_{v \preceq t}\alpha(v)\xi_v $$ for given weight functions $\alpha$ and $\sigma$ on $T$ and with $(\xi_v)_{v\in T}$ i.i.d. standard normal. In a first part we treat general trees and weights and derive necessary and sufficient conditions for the a.s. boundedness of $X$ in terms of compactness properties of $(T,d)$. Here $d$ is a special metric defined via $\alpha$ and $\sigma$, which, in general, is not comparable with the Dudley metric generated by $X$. In a second part we investigate the boundedness of $X$ for the binary tree and for homogeneous weights. Assuming some mild regularity assumptions about $\alpha$ we completely characterize weights $\alpha$ and $\sigma$ with $X$ being a.s. bounded.