Limit theorems on the convergence of infinite products for parametric stochastic operator systems (Q918032)

This is the fourth of a series of articles by these two authors in this journal. Let \(G_ 2(H)\) consist of all operators of the form \(I+A\) where I is the identity and A is a Hilbert-Schmidt operator on a Hilbert space H. Let \(\{\) Z(t,s): \(0\leq s\leq t\leq T\}\) be a family of \(G_ 2(H)\)- valued random variables. For \(\Delta =\{s=t_ 0<t_ 1<...<t_ n=t\}\), a partition of [s,t]\(\subseteq [0,T]\), let \(X_{\Delta}\) be the ordered product \(\prod^{n}_{i=1}Z(t_ i,t_{i-1}).\) It is shown that in some cases the limit X(t,s) of \(X_{\Delta}\) (as \(\max_{i}(t_ i-t_{i-1})\to 0)\) exists in a certain sense. Properties of the limit process \(\{\) X(t,s)\(\}\) are studied.