Extreme value statistics and return intervals in long-range correlated uniform deviates.

We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider (e.g., maximum relative to minimum) are such that the reference point from which the maximum is measured is itself a random quantity. We analytically calculate the limiting distributions for independent and identically distributed random variables, and use these as a reference point for correlated cases. The distributions are different from that of the maximum itself (i.e., a Weibull distribution), reflecting the fact that the distribution of the reference point either dominates over or convolves with the distribution of the maximum. The functional form of the limiting distributions is unaffected by correlations, although the convergence is slower. We show that our findings can be directly generalized to a wide class of stochastic processes. We also analyze return interval distributions, and compare them to recent conjectures of their functional form.

Maximal height statistics for 1/f(alpha) signals.

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 01 (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.

Extreme statistics for time series: distribution of the maximum relative to the initial value.

The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1/falpha power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRHI). The exact MRHI distribution is derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (random acceleration), and alpha=infinity (single sinusoidal mode). For other, intermediate values of alpha , the distribution is determined from simulations. We find that the MRHI distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all alpha. The two main distinguishing features of the MRHI distribution are the much larger weight for small relative heights and the divergence at zero height for alpha>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRHI distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.

Stationarity of extreme bursts in the solar wind.

Recent results have suggested that the statistics of bursts in the solar wind vary with solar cycle. Here, we show that this variation is basically absent if one considers extreme bursts. These are defined as threshold-exceeding events over the range of high thresholds for which their number decays as a power law. In particular, we find that the distribution of duration times and energies of extreme bursts in the solar wind ε parameter and similar observables are independent of the solar cycle and in this sense stationary, and show robust asymptotic power laws with exponents that are independent of the specific threshold. This is consistent with what has been observed for solar flares and, thus, provides evidence in favor of a link between solar flares and extreme bursts in the solar wind.

Order statistics of 1/fα signals.

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) = (x(k) -x(k) + 1) between the kth and (k+1)st largest values of the signal. The result d(k) k(-1), 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) =(x(1) - x(k))~ k(ß) is also examined. The exponent ß 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 ß(α = 2) = 1/2, ß(4) = 3/2, and ß(∞) = 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.

Renormalization-group theory for finite-size scaling in extreme statistics.

We present a renormalization-group (RG) approach to explain universal features of extreme statistics applied here to independent identically distributed variables. The outlines of the theory have been described in a previous paper, the main result being that finite-size shape corrections to the limit distribution can be obtained from a linearization of the RG transformation near a fixed point, leading to the computation of stable perturbations as eigenfunctions. Here we show details of the RG theory which exhibit remarkable similarities to the RG known in statistical physics. Besides the fixed points explaining universality, and the least stable eigendirections accounting for convergence rates and shape corrections, the similarities include marginally stable perturbations which turn out to be generic for the Fisher-Tippett-Gumbel class. Distribution functions containing unstable perturbations are also considered. We find that, after a transitory divergence, they return to the universal fixed line at the same or at a different point depending on the type of perturbation.

Finite-size scaling in extreme statistics.

We study the deviations from the limit distributions in extreme value statistics arising due to the finite size (FS) of data sets. A renormalization method is introduced for the case of independent, identically distributed (iid) variables, showing that the iid universality classes are subdivided according to the exponent of the FS convergence, which determines the leading order FS shape correction function as well. It is found that, for the correlated systems of subcritical percolation and 1/f;(alpha) stationary (alpha<1) noise, the iid shape correction compares favorably to simulations. Furthermore, for the strongly correlated regime (alpha>1) of 1/f;(alpha) noise, the shape correction is obtained in terms of the limit distribution itself.