Unconditional Convergence of Sub-Gaussian Random Series

Abstract

In this paper we explore the basic properties of sub-Gaussian random variables and random elements. We also present various notions of subgaussianity (weak, \({\mathbf{T}}\)- and \({\mathbf{F}}\)-subgaussianity) of random elements with values in general Banach spaces. It is shown that the covariance operator of \({\mathbf{T}}\)-subgaussian random element is Gaussian and some consequences of this result in spaces possessing certain geometric properties are noted. Moreover, the almost sure (a.s.) unconditional convergence of random series are considered and a sufficient condition of a.s. unconditional convergence of a random series of a special type with values in a Banach space with some geometric properties is proved. By the a.s. unconditional convergence of random series we understand the convergence of all rearrangements of the series on the same set of probability 1. With some effort, we prove one of the main results of the paper, which gives us a necessary condition for the a.s. unconditional convergence of random series of a special type in a general Banach space. For the proof, a lemma is used that establishes a connection between the moments of a random variable and which may be of independent interest.