Order statistics of 1/f^{\alpha} signals

Abstract

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=<x_k-x_{k+1}> 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=<x_1-x_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.