EXPONENTIAL INTEGRABILITY OF SUB-GAUSSIAN VECTORS

In this paper we define two classes of Banach space (B, ∥·∥)-valued random vectors called sub-Gaussian vectors and γ-sub-Gaussian vectors. The main purpose of this paper is to prove the exponential integrability of a sub-Gaussian vector X, that is, {Mathematical expression} for some ε>0, in the case where B=Lp. On the other hand, using the arguments of X. Fernique and M. Talagrand, we also show that the exponential integrability of a γ-sub-Gaussian vector in an arbitrary separable Banach space. These two definitions of sub-Gaussian vectors and γ-sub-Gaussian vectors are not comparable, and neither of these definitions is a necessary condition for the exponential integrability. We shall give illuminating examples. © 1990 Springer-Verlag.