Random Gaussian sums on trees

Abstract

Let $T$ be a tree with induced partial order $\preceq$. We investigate centered Gaussian processes $X=(X_t)_{t\in T}$ represented as $$ X_t=σ(t)\sum_{v \preceq t}α(v)ξ_v $$ for given weight functions $α$ and $σ$ on $T$ and with $(ξ_v)_{v\in T}$ 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 $α$ and $σ$, 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 $α$ we completely characterize weights $α$ and $σ$ with $X$ being a.s. bounded.