Boundedness of Gaussian random sums on trees

TL;DR

This paper establishes a precise criterion for the almost sure boundedness and uniform convergence of Gaussian sums on trees, linking the structure of the tree and the decay of coefficients.

Contribution

It provides a necessary and sufficient condition for Gaussian process boundedness on trees, extending understanding of Gaussian sums in complex hierarchical structures.

Findings

Derived a criterion for Gaussian process boundedness on trees

Connected tree structure with convergence properties of Gaussian sums

Established conditions for uniform convergence along geodesic rays

Abstract

Let $\mathcal{T}$ be a rooted tree endowed with the natural partial order $\preceq$. Let $(Z(v))_{v\in \mathcal{T}}$ be a sequence of independent standard Gaussian random variables and let $\alpha = (\alpha_k)_{k=1}^\infty$ be a sequence of real numbers with $\sum_{k=1}^\infty \alpha_k^2<\infty$. Set $\alpha_0 =0$ and define a Gaussian process on $\mathcal{T}$ in the following way: \[ G(\mathcal{T}, \alpha; v): = \sum_{u\preceq v} \alpha_{|u|} Z(u), \quad v \in \mathcal{T}, \] where $|u|$ denotes the graph distance between the vertex $u$ and the root vertex. Under mild assumptions on $\mathcal{T}$, we obtain a necessary and sufficient condition for the almost sure boundedness of the above Gaussian process. Our condition is also necessary and sufficient for the almost sure uniform convergence of the Gaussian process $G(\mathcal{T}, \alpha; v)$ along all rooted geodesic rays in $\mathcal{T}$.