A particular class of strong non-Markovian stochastic processes have been studied by using a characteristic functional technique previously reported. Exact results for all moments and the whole Kolmogorov hierarchy are presented. The asymptotic scaling of the non-Markovian stochastic process has been characterized in terms of the long-range correlated noise appearing in the correponding stochastic differential equation. A generalized Wiener process has therefore been completely characterized, its power spectrum and fractal dimensions have been studied and its possible connection with the q-statistics has been pointed out.