Supremum of a Process in Terms of Trees

详细信息    查看全文

• 作者：Olivier Guédon ; Artem Zvavitch
• 刊名：Lecture Notes in Mathematics
• 出版年：2003
• 出版时间：2003
• 年：2003
• 卷：1807
• 期：1
• 页码：p.136
• 全文大小：170 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Dynamical Systems and Ergodic Theory
  Mathematical Biology
  Partial Differential Equations
  Functional Analysis
  Abstract Harmonic Analysis
  Group Theory and Generalizations
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1617-9692

文摘

In this paper we study the quantity \mathbbE sup_{t ? T}X_t,\mathbb{E} \sup_{{t \in T}}X_{t}, where X_t is some random process. In the case of the Gaussian process, there is a natural sub-metric d defined on T. We find an upper bound in terms of labelled-covering trees of (T,d) and a lower bound in terms of packing trees (this uses the knowledge of packing numbers of subsets of T). The two quantities are proved to be equivalent via a general result concerning packing trees and labelled-covering trees of a metric space. Instead of using the majorizing measure theory, all the results involve the language of entropy numbers. Part of the results can be extended to some more general processes which satisfy some concentration inequality.