Electronic Journal of Probability

Random Gaussian Sums on Trees

Mikhail Lifshits and Werner Linde

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

This work is licensed under aCreative Commons Attribution 3.0 License.


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

Export citation


  • Bovier, Anton; Kurkova, Irina. Derrida's generalised random energy models. I. Models with finitely many hierarchies. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 4, 439–480.
  • Dudley, R. M. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Functional Analysis 1 1967 290–330.
  • Fernique, X. Regularité des trajectoires des fonctions aléatoires gaussiennes.(French) École d'Été de Probabilités de Saint-Flour, IV-1974, pp. 1–96. Lecture Notes in Math., Vol. 480, Springer, Berlin, 1975.
  • Fernique, X., Caractérisation de processus ? trajectoires majorées ou continues. Séminaire de Probabilitées, XII (Univ. Strasbourg, Strasbourg, 1976/1977), pp. 691–706, Lecture Notes in Math., vol. 649, Springer, Berlin, 1978.
  • Fernique, Xavier. Fonctions aléatoires gaussiennes, vecteurs aléatoires gaussiens.(French) [Gaussian random functions, Gaussian random vectors] Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1997. iv+217 pp. ISBN: 2-921120-28-3
  • Ledoux, Michel. Isoperimetry and Gaussian analysis. Lectures on probability theory and statistics (Saint-Flour, 1994), 165–294, Lecture Notes in Math., 1648, Springer, Berlin, 1996.
  • Lifshits, M. A. Gaussian random functions.Mathematics and its Applications, 322. Kluwer Academic Publishers, Dordrecht, 1995. xii+333 pp. ISBN: 0-7923-3385-3
  • Lifshits, M. A., Bounds for entropy numbers for some critical operators. To appear in Transactions of AMS (2010+). www.arxiv.org/abs/1002.1377
  • Lifshits, M. A. and Linde, W., Compactness properties of weighted summation operators on trees. Studia Math. 202 (2011), 17-47.
  • Lifshits, M. A. and Linde, W., Compactness properties of weighted summation operators on trees – the critical case. Preprint (2010).www.arxiv.org/abs/1009.2339
  • Pemantle, R., Search cost for a nearly optimal path in a binary tree. Ann. Appl. Probab. 19 (2009), 1273–1291.
  • Sudakov, V. N., Gaussian measures, Cauchy measures and $\epsilon$-entropy. Soviet Math. Dokl. 10 (1969), 310–313.
  • Talagrand, Michel. Regularity of Gaussian processes. Acta Math. 159 (1987), no. 1-2, 99–149.
  • Talagrand, M. New Gaussian estimates for enlarged balls. Geom. Funct. Anal. 3 (1993), no. 5, 502–526.
  • Talagrand, Michel. Majorizing measures: the generic chaining. Ann. Probab. 24 (1996), no. 3, 1049–1103.
  • Talagrand, Michel. The generic chaining. Upper and lower bounds of stochastic processes.Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005. viii+222 pp. ISBN: 3-540-24518-9