Order statistics of 1/f^α signals

Details Order statistics of 1/f^α signals Order statistics of 1/f^α signals

2011-09-25

arxiv.org/abs/1109.5360v1

Physics

Condensed Matter

Statistical Mechanics

8 pages, 5 figures

Scientific paper

Order statistics of periodic, Gaussian noise with 1/f^{\alpha} power spectrum is investigated. Using simulations and phenomenological arguments, we find three scaling regimes for the average gap d_k= between the k-th and (k+1)-st largest values of the signal. The result d_k ~ 1/k known for independent, identically distributed variables remains valid for 0<\alpha<1. Nontrivial, \alpha-dependent scaling exponents d_k ~ k^{(\alpha -3)/2} emerge for 1<\alpha<5 and, finally, \alpha-independent scaling, d_k ~ k is obtained for \alpha>5. The spectra of average ordered values \epsilon_k= ~ k^{\beta} is also examined. The exponent {\beta} is derived from the gap scaling as well as by relating \epsilon_k to the density of near extreme states. Known results for the density of near extreme states combined with scaling suggest that \beta(\alpha=2)=1/2, \beta(4)=3/2, and beta(infinity)=2 are exact values. We also show that parallels can be drawn between \epsilon_k and the quantum mechanical spectra of a particle in power-law potentials.

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