Characterizing measurability, distribution and weak convergence of random variables in a Banach space by total subsets of linear functionals

Abstract

Consider a generalized random variable X assuming values in a Banach space 4 with conjugate space 4*. For separable or reflexive 4 the measurability, probability distribution, and other properties of X are characterized in terms of a collection of real random variables {a*(X) : a* [set membership, variant] A} and their linear combinations, where A is a total subset of 4*, i.e., A distinguishes points of 4. Convergence in distribution of a sequence {Xn} is characterized in terms of uniform convergence of finite-dimensional distributions formed from {x*(Xn) : x* [set membership, variant] 4*} (for 4 separable) or from {a*(Xn) : a* [set membership, variant] A} (for 4 separable and reflexive). These results extend earlier ones known for the special cases A = 4* or 4 = C[0, 1]. The proofs are based on theorems of Banach, Krein, and Smulian characterizing the weak* closure of a convex set in 4*. An application to random Fourier series is given.Generalized random variables Banach space total subsets of linear functionals characterization of measurability probability distribution convergence in distribution weak convergence