Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet

✍ Scribed by T Dunker; W Linde; T Kühn; M.A Lifshits

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 146 KB
• Volume: 101
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1999.3354

✦ Synopsis

Let T d : L 2 ([0, 1] d ) Ä C([0, 1] d ) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k &1 (log k) d&1Â2 . From this we derive that the small ball probabilities of the Brownian sheet on [0, 1] d under the C([0, 1] d )-norm can be estimated from below by exp(&C= &2 |log =| 2d&1 ), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms.