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

We analyze the stochastic function $C_n\mathrm{}\left(i\right)\mathrm{}\equiv y\left(i\right)\mathrm{}-y_n\mathrm{}\left(i\right),$ where $y\left(i\right)\mathrm{}$ is a long-range correlated time series of length $N_{\max }$ and $y_n\mathrm{}\left(i\right)\mathrm{}\equiv \mathrm{}(1/n)\mathrm{}{\sum }_{k=0\mathrm{}}^{n-1}y\left(i-k\right)$ is the moving average with window n. We argue that $C_n\mathrm{}\left(i\right)\mathrm{}$ generates a stationary sequence of self-affine clusters $\mathcal{C}$ with length $\mathcal{l},$ lifetime $\tau ,$ and area s. The length and the area are related to the lifetime by the relationships $\mathcal{l}\sim {\tau }^{{\psi }_{\mathcal{l}}}$ and $s\sim {\tau }^{{\psi }_s},$ where ${\psi }_{\mathcal{l}}=1\mathrm{}$ and ${\psi }_s=1+H.\mathrm{}$ We also find that $\mathcal{l},$ $\tau ,$ and s are power law distributed with exponents depending on H: $P\left(\mathcal{l}\right)\sim {\mathcal{l}}^{-\alpha },$ $P\left(\tau \right)\sim {\tau }^{-\beta },$ and $P\left(s\right)\mathrm{}\sim s^{-\gamma },$ with $\alpha =\beta =2\mathrm{}-H$ 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 $N_{\max }{=2\mathrm{}}^{21}$ and n up to $2^{13}.$