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

We analyze the stochastic function Cₙ(i)≡y(i)-yₙ(i), where $y(i)$ is a long-range correlated time series of length Nₘₐₓ and yₙ(i)≡(1/n)∑ₖ₌₀^n-1y(i-k) is the moving average with window n. We argue that Cₙ(i) generates a stationary sequence of self-affine clusters C with length l, lifetime τ, and area s. The length and the area are related to the lifetime by the relationships l~τ^ψₗ and s~τ^ψₛ, where ψₗ=1 and ψₛ=1+H. We also find that l, τ, and s are power law distributed with exponents depending on H: P(l)~l^-α, P(τ)~τ^-β, and P(s)~s^-γ, with α=β=2-H and γ=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 Nₘₐₓ=2²¹ and n up to 2¹³.