2008/04/30 by Frank Aurzada, Mikhail Lifshits, Werner Linde · 1 citation
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Mathematical Approximation and Integration #Probability and Risk Models #Stochastic processes and financial applications #math.PR
paper · pdf · doi:10.3150/09-bej212
published as Bernoulli 2009, Vol. 15, No. 4, 1305-1334 · Published in at http://dx.doi.org/10.3150/09-BEJ212 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
openalex publication_date 2009/11/01 · arxiv created 2010/01/20 · arxiv updated 2010/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the relation between the small deviation problem for a symmetric α-stable random vector in a Banach space and the metric entropy properties of the operator generating it. This generalizes former results due to Li and Linde and to Aurzada. It is shown that this problem is related to the study of the entropy numbers of a certain random operator. In some cases, an interesting gap appears between the entropy of the original operator and that of the random operator generated by it. This phenomenon is studied thoroughly for diagonal operators. Basic ingredients here are techniques related to random partitions of the integers. The main result concerning metric entropy and small deviations allows us to determine or provide new estimates for the small deviation rate for several symmetric α-stable random processes, including unbounded Riemann–Liouville processes, weighted Riemann–Liouville processes and the (d-dimensional)α-stable sheet.