Extension of the Cameron-Martin theorem to random translations
成果类型：
Article
署名作者：
Fernique, X
署名单位：
Universites de Strasbourg Etablissements Associes; Universite de Strasbourg; Centre National de la Recherche Scientifique (CNRS); Universites de Strasbourg Etablissements Associes; Universite de Strasbourg
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/aop/1055425780
发表日期：
2003
页码：
1296-1304
关键词：
摘要：
Let G be a Gaussian vector taking its values in a separable Frechet space E. We denote by gamma its law and by (H, parallel to(.)parallel to) its reproducing Hilbert space. Moreover, let X be an E-valued random vector of law mu. In the first section, we prove that if mu is absolutely continuous relative to gamma, then there exist necessarily a Gaussian vector G' of law gamma and an H-valued random vector Z such that G' + Z has the law mu of X. This fact is a direct consequence of concentration properties of Gaussian vectors and, in some sense, it is an unexpected achievement of a part of the Cameron-Martin theorem. In the second section, using the classical Cameron-Martin theorem and rotation invariance properties of Gaussian probabilities, we show that, in many situations, such a condition is sufficient for mu being absolutely continuous relative to gamma.