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

Abstract

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.

Sharp moment and exponential tail estimates for U-statistics

“…It is known [28] that the Gψ norm of arbitrary r.v. ζ is complete equivalent to the its norm in Orlicz space…”

Modified Bernstein's inequality for sums of independent random variables

“…{ξ i } taking value in certain separable Banach space, may be finite dimensional R d . The Bernstein's inequality in the classical statement of this problem is investigated in many works: [31], [32], [33], [36] - [39] and so one.…”