{"paper":{"title":"Random Gaussian sums on trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Mikhail Lifshits, Werner Linde","submitted_at":"2010-12-13T10:27:54Z","abstract_excerpt":"Let $T$ be a tree with induced partial order $\\preceq$. We investigate centered Gaussian processes $X=(X_t)_{t\\in T}$ represented as $$\n  X_t=\\sigma(t)\\sum_{v \\preceq t}\\alpha(v)\\xi_v $$ 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 gene"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2683","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}