Random Gaussian Sums on Trees

Mikhail Lifshits (St. Petersburg State University)
Werner Linde (FSU Jena)

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.

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Pages: 739-763

Publication Date: April 12, 2011

DOI: 10.1214/EJP.v16-871

References

  1. 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. MR2070334 (2005h:82053)
  2. Dudley, R. M. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Functional Analysis 1 1967 290--330. MR0220340 (36 #3405)
  3. 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. MR0413238 (54 #1355)
  4. Fernique, X., CaractÈrisation de processus ? trajectoires majorÈes ou continues. SÈminaire de Probabilit\'es, XII (Univ. Strasbourg, Strasbourg, 1976/1977), pp. 691--706, Lecture Notes in Math., vol. 649, Springer, Berlin, 1978. MR0520032
  5. 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 MR1472975 (99f:60078)
  6. 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. MR1600888 (99h:60002)
  7. Lifshits, M. A. Gaussian random functions.Mathematics and its Applications, 322. Kluwer Academic Publishers, Dordrecht, 1995. xii+333 pp. ISBN: 0-7923-3385-3 MR1472736 (98k:60059)
  8. Lifshits, M. A., Bounds for entropy numbers for some critical operators. To appear in Transactions of AMS (2010+). www.arxiv.org/abs/1002.1377
  9. Lifshits, M. A. and Linde, W., Compactness properties of weighted summation operators on trees. Studia Math. 202 (2011), 17-47.
  10. 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
  11. Pemantle, R., Search cost for a nearly optimal path in a binary tree. Ann. Appl. Probab. 19 (2009), 1273--1291. MR2538070
  12. Sudakov, V. N., Gaussian measures, Cauchy measures and $\epsilon$-entropy. Soviet Math. Dokl. 10 (1969), 310--313.
  13. Talagrand, Michel. Regularity of Gaussian processes. Acta Math. 159 (1987), no. 1-2, 99--149. MR0906527 (89b:60106)
  14. Talagrand, M. New Gaussian estimates for enlarged balls. Geom. Funct. Anal. 3 (1993), no. 5, 502--526. MR1233864 (94k:60004)
  15. Talagrand, Michel. Majorizing measures: the generic chaining. Ann. Probab. 24 (1996), no. 3, 1049--1103. MR1411488 (97k:60097)
  16. 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 MR2133757 (2006b:60006)
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