Random Gaussian sums on trees
Mikhail Lifshits, Werner Linde

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 be a tree with induced partial order . We investigate centered Gaussian processes represented as for given weight functions and on and with 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 in terms of compactness properties of . Here is a special metric defined via and , which, in general, is not comparable with the Dudley metric generated by . In a second part we investigate the boundedness of for the binary tree and for homogeneous weights. Assuming some mild regularity assumptions about we completely characterize weights and with being a.s. bounded.
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Taxonomy
TopicsGeometry and complex manifolds · Stochastic processes and statistical mechanics · Geometric Analysis and Curvature Flows
