Extension du théorème de Cameron--Martin aux translations aléatoires

• Xavier Fernique ^[1]
  1. [1] Université Louis Pasteur et C.N.R.S.
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 3, 2003, págs. 1296-1304
• Idioma: inglés
• DOI: 10.1214/aop/1055425780
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• Resumen

  • Let G be a Gaussian vector taking its values in a separable Fréchet space E. We denote by γ its law and by (H,∥⋅∥) its reproducing Hilbert space. Moreover, let X be an E-valued random vector of law μ. In the first section, we prove that if μ is absolutely continuous relative to γ, then there exist necessarily a Gaussian vector G′ of law γ and an H-valued random vector Z such that G′+Z has the law μ 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 μ being absolutely continuous relative to γ.