In an earlier paper [3] the authors gave a characterization of the Wiener process by the property that a stochastic integral\(\int_A^B {a\left( t \right)dX\left( t \right)} \) (taken in the sense of convergence in the quadratic mean) has the same distribution as α[X(t+1)−X(t)]. The assumption that the integral was defined in the sense of convergence in the quadratic mean necessitated a number of hypotheses; for instance one had to suppose that the process was of the second order and that its mean value function and its covariance function were of bounded variation.