SUB-GAUSSIAN PROPERTY OF POSITIVE GENERALIZED WIENER FUNCTIONS
| Autor: | Fukuda, Ryoji |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 1994 |
| Zdroj: | Kyushu Journal of Mathematics. 48(1):201-220 |
| ISSN: | 1340-6116 |
| Popis: | A real random variable X is sub-Gaussian iff there exist $ K>0 $ such that $ E[ exp ( lambda X) ] leqq exp (K^2 \times lambda^2 2) $ for any $ lambda in R $. J-P. Kahane [10] proved that a real random variable $ X $ is sub-Gaussian if and only if $ E[X]=0 $ and $ E[ exp ( \varepsilon X^2)]< infty $ for some $ \varepsilon>0 $. A probability measure $ mu $ on a Banach space $ B $ is said to be sub-Gaussian iff there exists $ C>0 $ such that $ int_B exp()mu(dx) leqq exp( \fraq{C^2}{2} int^2 mu(dx)) < infty $ for any $ y in B^* $. (1) A Gaussian measure and the probability measure induced by a Rademacher series are typical examples of sub-Gaussian measures, and for these two probability measures, $ exp( \varepsilon Arrowvert x^2 Arrowvert ) $ is integrable for some $ epsilon>0 $ ([3], [12]). We call this integrability the exponential square integrability. When $ B=L_p $ $ (p geqq 1) $, (1) is a sufficient condition for the exponential square integrability, but not necessary even if $ B $ is a Hilbert space ([4]). For a sub-Gaussian measure $ mu $, the $ L_p( mu ) $ topologies $ (0;y in B^*} $. To show that considerably many probability measures satisfy such a remarkable property, we shall propve the sub-Gaussian property of probability measures for two types. One is a probability measure identified with a positive generalized Wiener function (see H. Sugita [17]), and the other is a probability measure which is absolutely continuous with respect to the probability measure induced by a random Fourier series. The former is exponentially square integrable [17], and so is the latter under certain additional conditions. |
| Databáze: | OpenAIRE |
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