ABSTRACT

Let X be a real separable Banach space with the dual space, X∗, the unit ball BX , and the unit sphere SX . By definition, a random vector X with values in X is a strongly measurable map from the probability space (Ω,Σ,P) (always sufficiently rich) into X equipped with the Borel σ -algebra BX . The set of all random vectors in X will be denoted L0(Ω,Σ,P;X) or, simply, L0(X), and will be equipped with the topology of convergence in probability which is determined by the family of gauges,

Jα(X,P) := inf { c : P(‖X‖ > c) ≤ α}, α ∈ (0, 1). (1.1.1)

Random vectors on product spaces (Ω1 × Ω2,Σ1 × Σ2,P1 × P2) satisfy the following Fubini inequality,1

Jγ(Jδ(X,P1),P2) ≤ Jα(Jβ(X,P2),P1), (1.1.2)

whenever α + β ≤ γδ (see, also, (1.3.1(b)). 1

in

By Lp(Ω,Σ,P;X), or, simply, Lp(X), 0 < p ≤ ∞, we shall denote the space of random vectors X in X for which E‖X‖p :=∫ Ω ‖X(ω)‖pP(dω) <∞, if p <∞, and ess sup‖X‖ <∞, if p =∞,

equipped with the corresponding topologies and quasi-norms. lp will denote the analogous spaces on the set of positive integers N, and s will denote the space of real sequences with finitely many non-zero terms.