Electronic Journal of Probability

Random Gaussian Sums on Trees

Mikhail Lifshits and Werner Linde

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Abstract

Let $T$ be a tree with induced partial order. We investigate a centered Gaussian process $X$ indexed by $T$ and generated by weight functions. 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 by the weights, 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. Assuming some mild regularity assumptions about on weight, we completely characterize homogeneous weights with $X$ being a.s. bounded.

Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 24, 739-763.

Dates
Accepted: 12 April 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464820193

Digital Object Identifier
doi:10.1214/EJP.v16-871

Mathematical Reviews number (MathSciNet)
MR2786646

Zentralblatt MATH identifier
1226.60053

Subjects
Primary: 60G15: Gaussian processes
Secondary: 06A06: Partial order, general 05C05: Trees

Keywords
Gaussian processes processes indexed by trees bounded processes summation on trees metric entropy

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Lifshits, Mikhail; Linde, Werner. Random Gaussian Sums on Trees. Electron. J. Probab. 16 (2011), paper no. 24, 739--763. doi:10.1214/EJP.v16-871. https://projecteuclid.org/euclid.ejp/1464820193


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