SUB-GAUSSIAN PROPERTY OF POSITIVE GENERALIZED WIENER FUNCTIONS
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概要
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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^22) $ 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(<y,x>)\mu(dx) \leqq \exp( \fraq{C^2}{2} \int<y,x>^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<p<\infty) $ and the $ L_0(μ) $ topology coincide on the family ${<y,x>;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.
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