Random Gaussian Sums on Trees
| Autor: | Mikhail Lifshits, Werner Linde |
|---|---|
| Rok vydání: | 2011 |
| Předmět: |
Statistics and Probability
processes indexed by trees Gaussian Gaussian processes Combinatorics summation on trees 05C05 symbols.namesake FOS: Mathematics 60G15 06A06 05C05 Gaussian process Mathematics bounded processes Binary tree Probability (math.PR) Order (ring theory) metric entropy Compact space 06A06 Bounded function 60G15 Metric (mathematics) symbols Tree (set theory) Statistics Probability and Uncertainty Mathematics - Probability |
| Zdroj: | Electron. J. Probab. 16 (2011), 739-763 Scopus-Elsevier |
| ISSN: | 1083-6489 |
| DOI: | 10.1214/ejp.v16-871 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|