Polynomial convergence of iterations of certain random operators in Hilbert space

We prove the polynomial convergence rate of the average of the sequence which is explicitly determined only by the regularity of the initial state, Theorem 1 and 2. For convergence of the second moment, under a condition on the regularity of the random distribution (Assumption (A)), the convergence rate remains the same, Theorem 3. In another words, under (A), the regularity of the random sequence only affects the coefficient not the order of the polynomial convergence. Additionally we demonstrate almost sure convergence of the sequence, Theorem 4.