Analysis of clusters formed by the moving average of a long-range correlated time series.

Abstract

We analyze the stochastic function C(n)(i) identical with y(i)-y(n)(i), where y(i) is a long-range correlated time series of length N(max) and y(n)(i) identical with (1/n) Sigma(n-1)(k=0)y(i-k) is the moving average with window n. We argue that C(n)(i) generates a stationary sequence of self-affine clusters C with length l, lifetime tau, and area s. The length and the area are related to the lifetime by the relationships l approximately tau(psi(l)) and s approximately tau(psi(s)), where psi(l)=1 and psi(s)=1+H. We also find that l, tau, and s are power law distributed with exponents depending on H: P(l) approximately l(-alpha), P(tau) approximately tau(-beta), and P(s) approximately s(-gamma), with alpha=beta=2-H and gamma=2/(1+H). These predictions are tested by extensive simulations on series generated by the midpoint displacement algorithm of assigned Hurst exponent H (ranging from 0.05 to 0.95) of length up to N(max)=2(21) and n up to 2(13).