Statistical Properties of Multivariate Fractional Noise Processes

Fractional noise processes belong to a class of stochastic processes that lie outside the Brownian domain of attraction. They are characterized by infinite memories and a parameter h that is the asymptotic slope of log ( R / S ) versus log N , where R is the range of cumulative departures from the sample mean, s is the sample standard deviation, and N is the sample size. The parameter h may be used to generate synthetic flows whose means, variances, skewnesses, lag one serial correlations, lag zero cross correlations, and h values are equal to those for historical flow sequences. Historical flow sequences yield values of h ≠ ½. Although Markovian processes, which belong to the Brownian domain, can preserve the values of the historical moments in the synthetic sequences, these processes generate sequences in which h tends to the value ½ as the sequence lengths tend to infinity.