Convergence of infinite products of independent random almost-linear operators (Q1091676)

Let \(\{X_ k\}\) be a sequence of almost linear random independent operators in a real separable Hilbert space such that \(\sup_{| x| \leq 1}M| X_ k(x)|^ 2<\infty\). The main result (Theorem 2) states: If \(MX_ k=0\), \(\sum^{n}_{k=1}M\| X_ k\|^ 2<\infty\) and \((E+X_ k)\) are invertible, then the products \(\prod^{n}_{k=1}(E+X_ k)\) and \(\prod^{n}_{k=1}(E+X_ k)^{- 1}\) converge strongly strongly with probability 1.