Convergence of Random Bounded Linear Operators in the Skorokhod Space

Abstract

Let Y a separable Banach space, and Β(Y) the space of bounded linear operators Y → Y. Let D the space of maps S : R → Β(Y) such that Sf ∈ D(R ,Y) ∀f ∈ Y, where D is the Skorokhod space of right-continuous functions with left-hand limits. Say we are given a sequence {Sn}n∈N of random maps Ω → D - where (Ω,F,P) is a probability space - and that we understand well the convergence of Sn f for each f ∈ Y . What can we say about the convergence of Sn ? In the following, we introduce the notion of D−valued random variable and prove almost sure and weak convergence results for sequences of such random variables.