Second-order moving average and scaling of stochastic time series

Long-range correlation properties of stochastic time series y(i) have been investigated by introducing the function $\sigma_{MA}^2=\frac{1}{N_{max}-n}\sum_{i=n}^{N_{max}} [y(i)-\widetilde{y}_n(i)]^2$, where $\widetilde{y}_n(i)$ is the moving average of y(i), defined as $1/n \sum_{k=0}^{n-1} y(i-k)$, n the moving average window and Nmax is the dimension of the stochastic series. It is shown that, using an appropriate computational procedure, the function $\sigma_{MA}$ varies as nH where H is the Hurst exponent of the series. A comparison of the power-law exponents obtained using respectively the function $\sigma_{MA}$ and the Detrended Fluctuation Analysis has been also carried out. Interesting features denoting the existence of a relationship between the scaling properties of the noisy process and the moving average filtering technique have been evidenced