SUB-GAUSSIAN PROPERTY OF POSITIVE GENERALIZED WIENER FUNCTIONS

Abstract

A real random variable X is sub-Gaussian iff there exist K>0 such thatE[exp(λX)]≤exp(??) for any λ ∈ RJ-P. Kahane [10] proved that a real random variable X is sub-Gaussian if and only if E[X]=0 and E[exp(eX2)] 0. A probability measure μ on a Banach space B is said to be sub-Gaussian iff there exists C>0 such that∫Bexp(y, x>)μ(dx)≤exp(??∫y, x>2μ(dx)) 0 ([3], [12]). We call this integrability the exponential square integrability. When B = L P (p≥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 μ, the Lp (μ) topologies (0 ; y ∈ 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.