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

Generating discrete-time constrained random walks and Lévy flights

We introduce a method to exactly generate bridge trajectories for discrete-time random walks, with arbitrary jump distributions, that are constrained to initially start at the origin and return to the origin after a fixed time. The method is based on an effective jump distribution that implicitly accounts for the bridge constraint. It is illustrated on various jump distributions and is shown to be very efficient in practice. In addition, we show how to generalize the method to other types of constrained random walks such as generalized bridges, excursions, and meanders.

Records in fractal stochastic processes

The record statistics in stationary and non-stationary fractal time series is studied extensively. By calculating various concepts in record dynamics, we find some interesting results. In stationary fractional Gaussian noises, we observe a universal behavior for the whole range of Hurst exponents. However, for non-stationary fractional Brownian motions, the record dynamics is crucially dependent on the memory, which plays the role of a non-stationarity index, here. Indeed, the deviation from the results of the stationary case increases by increasing the Hurst exponent in fractional Brownian motions. We demonstrate that the memory governs the dynamics of the records as long as it causes non-stationarity in fractal stochastic processes; otherwise, it has no impact on the record statistics.

Near-extreme statistics of Brownian motion

We study the statistics of near-extreme events of Brownian motion (BM) on the time interval [0,t]. We focus on the density of states near the maximum, ρ(r,t), which is the amount of time spent by the process at a distance r from the maximum. We develop a path integral approach to study functionals of the maximum of BM, which allows us to study the full probability density function of ρ(r,t) and obtain an explicit expression for the moments <[ρ(r,t)]k> for arbitrary integer k. We also study near extremes of constrained BM, like the Brownian bridge. Finally we also present numerical simulations to check our analytical results.

Universal fluctuations and extreme statistics of avalanches near the depinning transition

We derive exact predictions for universal scaling exponents and scaling functions associated with the statistics of maximum velocities v(m) during avalanches described by the mean-field theory of the interface depinning transition. In particular, we find a robust power-law regime in the statistics of maximum events that can explain the observed distribution of the peak amplitudes in acoustic emission experiments of crystal plasticity. Our results are expected to be broadly applicable to a broad range of systems in the mean-field interface depinning universality class, ranging from magnets to earthquakes.

Positive feedback produces broad distributions in maximum activation attained within a narrow time window in stochastic biochemical reactions

How do single cell fate decisions induced by activation of key signaling proteins above threshold concentrations within a time interval are affected by stochastic fluctuations in biochemical reactions? We address this question using minimal models of stochastic chemical reactions commonly found in cell signaling and gene regulatory systems. Employing exact solutions and semi-analytical methods we calculate distributions of the maximum value (N) of activated species concentrations (P(max)(N)) and the time (t) taken to reach the maximum value (P(max)(t)) within a time interval in the minimal models. We find, the presence of positive feedback interactions make P(max)(N) more spread out with a higher "peakedness" in P(max)(t). Thus positive feedback interactions may help single cells to respond sensitively to a stimulus when cell decision processes require upregulation of activated forms of key proteins to a threshold number within a time window.

Distribution of maximum velocities in avalanches near the depinning transition

We report exact predictions for universal scaling exponents and scaling functions associated with the distribution of the maximum collective avalanche propagation velocities v(m) in the mean field theory of the interface depinning transition. We derive the extreme value distribution P(v(m)|T) for the maximum velocities in avalanches of fixed duration T and verify the results by numerical simulation near the critical point. We find that the tail of the distribution of maximum velocity for an arbitrary avalanche duration, v(m), scales as P(v(m))~v(m)(-2) for large v(m). These results account for the observed power-law distribution of the maximum amplitudes in acoustic emission experiments of crystal plasticity and are also broadly applicable to other systems in the mean-field interface depinning universality class, ranging from magnets to earthquakes.

Universal order statistics of random walks

We study analytically the order statistics of a time series generated by the positions of a symmetric random walk of n steps with step lengths of finite variance σ(2). We show that the statistics of the gap d(k,n) = M(k,n)-M(k+1,n) between the kth and the (k+1)th maximum of the time series becomes stationary, i.e., independent of n as n → ∞ and exhibits a rich, universal behavior. The mean stationary gap exhibits a universal algebraic decay for large k, ~d(k,∞)-/σ 1/sqrt[2πk], independent of the details of the jump distribution. Moreover, the probability density (pdf) of the stationary gap exhibits scaling, Pr(d(k,∞) = δ) ~/= (sqrt[k]/σ)P(δsqrt[k]/σ), in the regime δ~ (d(k,∞)). The scaling function P(x) is universal and has an unexpected power law tail, P(x) ~ x(-4) for large x. For δ>> (d(k,∞)) the scaling breaks down and the pdf gets cut off in a nonuniversal way. Consequently, the moments of the gap exhibit an unusual multiscaling behavior.

Critical polyelectrolyte adsorption under confinement: planar slit, cylindrical pore, and spherical cavity

We explore the properties of adsorption of flexible polyelectrolyte chains in confined spaces between the oppositely charged surfaces in three basic geometries. A method of approximate uniformly valid solutions for the Green function equation for the eigenfunctions of polymer density distributions is developed to rationalize the critical adsorption conditions. The same approach was implemented in our recent study for the "inverse" problem of polyelectrolyte adsorption onto a planar surface, and on the outer surface of rod-like and spherical obstacles. For the three adsorption geometries investigated, the theory yields simple scaling relations for the minimal surface charge density that triggers the chain adsorption, as a function of the Debye screening length and surface curvature. The encapsulation of polyelectrolytes is governed by interplay of the electrostatic attraction energy toward the adsorbing surface and entropic repulsion of the chain squeezed into a thin slit or small cavities. Under the conditions of surface-mediated confinement, substantially larger polymer linear charge densities are required to adsorb a polyelectrolyte inside a charged spherical cavity, relative to a cylindrical pore and to a planar slit (at the same interfacial surface charge density). Possible biological implications are discussed briefly in the end.

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(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 β(∞) = 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.

Distribution of the time at which N vicious walkers reach their maximal height

We study the extreme statistics of N nonintersecting Brownian motions (vicious walkers) over a unit time interval in one dimension. Using path-integral techniques we compute exactly the joint distribution of the maximum M and of the time τ(M) at which this maximum is reached. We focus in particular on nonintersecting Brownian bridges ("watermelons without wall") and nonintersecting Brownian excursions ("watermelons with a wall"). We discuss in detail the relationships between such vicious walkers models in watermelon configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium (DPRM) with one free end point on the other hand. We also check our results using numerical simulations of Dyson's Brownian motion and confront them with numerical simulations of the polynuclear growth model (PNG) and of a model of DPRM on a discrete lattice. Some of the results presented here were announced in a recent letter [J. Rambeau and G. Schehr, Europhys. Lett. 91, 60006 (2010)].

Distribution of first-return times in correlated stationary signals

We present an analytical expression for the first return time (FRT) probability density function of a stationary correlated signal. Precisely, we start by considering a stationary discrete-time Ornstein-Uhlenbeck (OU) process with exponential decaying correlation function. The first return time distribution for this process is derived by adopting a well-known formalism typically used in the study of the FRT statistics for nonstationary diffusive processes. Then, by a subordination approach, we treat the case of a stationary process with power-law tail correlation function and diverging correlation time. We numerically test our findings, obtaining in both cases a good agreement with the analytical results. We notice that neither in the standard OU nor in the subordinated case a simple form of waiting time statistics, like stretched-exponential or similar, can be obtained while it is apparent that long time transient may shadow the final asymptotic behavior.

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.

Exact distribution of the maximal height of p vicious walkers

Using path-integral techniques, we compute exactly the distribution of the maximal height Hp of p nonintersecting Brownian walkers over a unit time interval in one dimension, both for excursions p watermelons with a wall, and bridges p watermelons without a wall, for all integer p>or=1. For large p, we show that <Hp> approximately square root 2p (excursions) whereas <Hp> approximately square root p (bridges). Our exact results prove that previous numerical experiments only measured the preasymptotic behaviors and not the correct asymptotic ones. In addition, our method establishes a physical connection between vicious walkers and random matrix theory.

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.