Banach spaces characterization of random vectors with exponential decreasing tails of distribution

We present in this paper the Banach space representation for the set of random finite-dimensional vectors with exponential decreasing tails of distributions. We show that there are at last three types of these multidimensional Banach spaces, i.e. which can completely describe the random vectors with exponential decreasing tails of distributions: exponential Orlicz spaces, Young spaces and Grand Lebesgue spaces. We discuss in the last section the possible applications of obtained results.