Random Gaussian sums on trees

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.