Order statistics of 1/f^α signals

Abstract

Order statistics of periodic, Gaussian noise with 1/f^α 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<α<1. Nontrivial, α-dependent scaling exponents d_k ~ k^{(α-3)/2} emerge for 1<α<5 and, finally, α-independent scaling, d_k ~ k is obtained for α>5. The spectra of average ordered values ε_k= ~ k^β is also examined. The exponent β is derived from the gap scaling as well as by relating ε_k to the density of near extreme states. Known results for the density of near extreme states combined with scaling suggest that β(α=2)=1/2, β(4)=3/2, and beta(infinity)=2 are exact values. We also show that parallels can be drawn between ε_k and the quantum mechanical spectra of a particle in power-law potentials.