On the sample continuity of random mappings (Q1808895)

Let \(\Phi=\{\Phi_x\}\) be a random mapping from \(X\) to \(Y\), i.e. a family of \(Y\)-valued random variables indexed by \(X\) (\((X,d)\) is a separable metric space and \(Y\) a separable Banach space). The author shows that a sufficient condition, of the type of Kolmogorov's criterion, for the sample continuity of \(\Phi\), which is established when \(X\) is finite-dimensional, fails in the case when \(X\) is a bounded set of an infinite-dimensional Hilbert space. Then he solves the problem of determining sufficient conditions for \(\Phi\) to be continuous in the infinite-dimensional setting. For this he defines the operator \(T:L_2(\Omega)\rightarrow C(X,Y)\) associated to \(\Phi\) as the Bochner integral: \(T(\xi)x=\int_{\Omega}\xi(\omega)\Phi_x(\omega)dP(\omega)\). The random mapping verifies (i) \(E||\Phi_x||^2<\infty\) and (ii) \((E||\Phi x_1-\Phi x_2||^2)^{1/2}\leq C d^{\delta}(x_1,x_2)\). In Theorem 2.8 he obtains that \(\Phi\) is sample continuous if \(T\) is 2-summing. Afterwards he considers random operators obtaining some results about their sample continuity.