Exponential integrability of sub-Gaussian vectors

Summary In this paper we define two classes of Banach space (B, ·)-valued random vectors called sub-Gaussian vectors and -sub-Gaussian vectors. The main purpose of this paper is to prove the exponential integrability of a sub-Gaussian vectorX, that is, for some >0, in the case whereB=L _p . On the other hand, using the arguments ofX. Fernique and M. Talagrand, we also show that the exponential integrability of a -sub-Gaussian vector in an arbitrary separable Banach space.These two definitions of sub-Gaussian vectors and -sub-Gaussian vectors are not comparable, and neither of these definitions is a necessary condition for the exponential integrability. We shall give illuminating examples.