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#### Order statistics of 1/f(alpha) signals

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##### Citation

Moloney, N. R., Ozogany, K., & Racz, Z. (2011). Order statistics of 1/f(alpha)
signals.* Physical Review E,* *84*(6): 061101.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0029-8BAA-1

##### 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 kth and (k + 1)st largest values of the signal. The result d(k) similar to k(-1), known for independent, identically distributed variables, remains valid for 0 <= alpha < 1. Nontrivial, alpha-dependent scaling exponents, d(k) similar to k((alpha-3)/2), emerge for 1 < alpha < 5, and, finally, alpha-independent scaling, d(k) similar to k, is obtained for alpha > 5. The spectra of average ordered values epsilon(k) = (x(1) - x(k)) similar to 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.