Maximal height statistics for 1/f^alpha signals

TL;DR

This paper investigates the statistical behavior of the maximum relative height of Gaussian signals with 1/f^alpha spectra, revealing different convergence behaviors and deriving analytical formulas for specific cases.

Contribution

It provides new analytical and numerical insights into the maximal height distribution of 1/f^alpha signals, including trace formulas and asymptotic behaviors for various alpha regimes.

Findings

Limiting distribution approaches Gumbel distribution slowly for 0<alpha<1.

Distribution converges rapidly for alpha>1.

Analytical formulas derived for alpha=2n even integers.

Abstract

Numerical and analytical results are presented for the maximal relative height distribution of stationary periodic Gaussian signals (one dimensional interfaces) displaying a 1/f^alpha power spectrum. For 0<alpha<1 (regime of decaying correlations), we observe that the mathematically established limiting distribution (Fisher-Tippett-Gumbel distribution) is approached extremely slowly as the sample size increases. The convergence is rapid for alpha>1 (regime of strong correlations) and a highly accurate picture gallery of distribution functions can be constructed numerically. Analytical results can be obtained in the limit alpha -> infinity and, for large alpha, by perturbation expansion. Furthermore, using path integral techniques we derive a trace formula for the distribution function, valid for alpha=2n even integer. From the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha > 1 is found to be in agreement with simulations. Comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions.