Arcsine Law Research Articles

Financial Data Analysis by means of Coupled Continuous-Time Random Walk in Rachev-Rűschendorf Model

We adapt the continuous-time random walk formalism to describe assetprice evolution. We expand the idea proposed by Rachev and Rusc˜ hendorfwho analyzed the binomial pricing model in the discrete time with random-ization of the number of price changes. As a result, in the framework of theproposed model we obtain a mixture of the Gaussian and a generalized arc-sine laws as the limiting distribution of log-returns. Moreover, we derive anEuropean-call-option price that is an extension of the Black{Scholes formula.We apply the obtained theoretical results to model actual flnancial data andtry to show that the continuous-time random walk ofiers alternative tools todeal with several complex issues of flnancial markets.PACS numbers: 89.65.Gh, 05.40.Fb

On Functional Versions of the Arc-Sine Law

Let X1 ,X 2 ,... be a sequence of random variables. Let Sk = X1 +···+ Xk and assume that Sk/bk converges in distribution for some numerical sequence (bk). We study the weak convergence of the random processes {Λn(z), z ∈ R}, where Λn(z) = 1 n �

Universality of the REM for Dynamics of Mean-Field Spin Glasses

We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ β ,p > 0, such that for all exponential time scales, exp(γ N), with γ < γ β ,p , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β 2 < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.

Generalized Arcsine Law and Stable Law in an Infinite Measure Dynamical System

Limit theorems for the time average of some observation functions in an infinite measure dynamical system are studied. It is known that intermittent phenomena, such as the Rayleigh-Benard convection and Belousov-Zhabotinsky reaction, are described by infinite measure dynamical systems.We show that the time average of the observation function which is not the $L^1(m)$ function, whose average with respect to the invariant measure $m$ is finite, converges to the generalized arcsine distribution. This result leads to the novel view that the correlation function is intrinsically random and does not decay. Moreover, it is also numerically shown that the time average of the observation function converges to the stable distribution when the observation function has the infinite mean.

Complex Hermite polynomials: from the semi-circular law to the circular law

We study asymptotics of orthogonal polynomial measures of the form |HN| 2 d∞ where HN are real or complex Hermite polynomials with re- spect to the Gaussian measure ∞. By means of dierential equations on Laplace transforms, interpolation between the (real) arcsine law and the (complex) uniform distribution on the circle is emphasized. Suitable aver- ages by an independent uniform law give rise to the limiting semi-circular and circular laws of Hermitian and non-Hermitian Gaussian random matrix models. The intermediate regime between strong and weak non-Hermiticity is clearly identified on the limiting dierential equation by means of an addi- tional normal variable in the vertical direction.

Testing a priority-based queue model with Linux command histories

We study human dynamics by analyzing Linux history files. The goodness-of-fit test shows that most of the collected datasets belong to the universality class suggested in the literature by a variable-length queuing process based on priority. In order to check the validity of this model, we design two tests based on mutual information between time intervals and a mathematical relationship known as the arcsine law. Since the previously suggested queuing process fails to pass these tests, the result suggests that the modelling of human dynamics should properly consider the statistical dependency in the temporal dimension.

A limit theorem for occupation times of Lamperti’s stochastic processes

The arcsine law for random walks on the line is well known and it was extended greatly by Lamperti for a class of discrete-time stochastic processes. In the present paper we treat its extreme case where the excursion intervals have very heavy tail probabilities. The result is a refinement of Lamperti’s theorem. A functional limit theorem is also discussed.

Infinite measure preserving transformations with compact first regeneration

We study ergodic infinite measure preserving transformations T possessing reference sets of finite measure for which the set of densities of the conditional distributions given a first return (or entrance) at time n is precompact in a suitable function space. Assuming regular variation of wandering rates, we establish versions of the Darling-Kac theorem and the arcsine laws for waiting times and for occupation times which apply to transformations with indifferent orbits and to random walks driven by Gibbs-Markov maps.

Asymptotic arc-sine laws for finite-dimensional interacting diffusions

We consider finite-dimensional interacting diffusions which are defined by adding a linear drift term to independent one dimensional diffusions. For these processes we prove that the distribution of the occupation time at the first quadrant converges to a generalized arc-sine law.

Arcsine Law for Branching Processes in a Random Environment and Galton–Watson Processes

Two closely related results are established. A given critical branching process in a random environment attains a high level and spends at that level a part of its life obeying the arcsine law. If a critical Galton–Watson process survives up to a distant moment, then the ratio of the total number of individuals of the future generations to the total number of individuals ever born in the process obeys the arcsine law.

The arcsine law as a universal aging scheme for trap models

AbstractWe give a general proof of aging for trap models using the arcsine law for stable subordinators. This proof is based on abstract conditions on the potential theory of the underlying graph and on the randomness of the trapping landscape. We apply this proof to aging for trap models on large, two‐dimensional tori and for trap dynamics of the random energy model on a broad range of time scales. © 2006 Wiley Periodicals, Inc.

Random walk to a nonergodic equilibrium concept

Random walk models, such as the trap model, continuous time random walks, and comb models, exhibit weak ergodicity breaking, when the average waiting time is infinite. The open question is, what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory for such systems? In this paper a nonergodic equilibrium concept is investigated, for a continuous time random walk model in a potential field. In particular we show that in the nonergodic phase the distribution of the occupation time of the particle in a finite region of space approaches U- or W-shaped distributions related to the arcsine law. We show that when conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the nonergodic dynamics and canonical statistical mechanics. In the ergodic phase the distribution function of the occupation times approaches a delta function centered on the value predicted based on standard Boltzmann-Gibbs statistics. The relation of our work to single-molecule experiments is briefly discussed.

Some random times and martingales associated with BES0(δ) processes (0

In this paper, we study Bessel processes of dimension $\delta\equiv2(1-\mu)$, with $0<\delta<2$, and some related martingales and random times. Our approach is based on martingale techniques and the general theory of stochastic processes (unlike the usual approach based on excursion theory), although for $0<\delta<1$, these processes are even not semimartingales. The last time before 1 when a Bessel process hits 0, called $g_{\mu}$, plays a key role in our study: we characterize its conditional distribution and extend Paul L\'{e}vy's arc sine law and a related result of Jeulin about the standard Brownian Motion. We also introduce some remarkable families of martingales related to the Bessel process, thus obtaining in some cases a one parameter extension of some results of Az\'{e}ma and Yor in the Brownian setting: martingales which have the same set of zeros as the Bessel process and which satisfy the stopping theorem for $g_{\mu}$, a one parameter extension of Az\'{e}ma's second martingale, etc. Throughout our study, the local time of the Bessel process also plays a central role and we shall establish some of its elementary properties.

Brownian representation of a class of Lévy processes and its application to occupation times of diffusion processes

It is well known that a class of subordinators can be represented using the local time of Brownian motions. An extension of such a representation is given for a class of Lévy processes which are not necessarily of bounded variation. This class can be characterized by the complete monotonicity of the Lévy measures. The asymptotic behavior of such processes is also discussed and the results are applied to the generalized arc-sine law, an occupation time problem on the positive side for one-dimensional diffusion processes.

Occupation times and ergodicity breaking in biased continuous time randomwalks

Continuous time random walk (CTRW) models are widely used to model diffusion incondensed matter. There are two classes of such models, distinguished by the convergenceor divergence of the mean waiting time. Systems with finite average sojourn time areergodic and thus Boltzmann–Gibbs statistics can be applied. We investigate the statisticalproperties of CTRW models with infinite average sojourn time; in particular, theoccupation time probability density function is obtained. It is shown that in thenon-ergodic phase the distribution of the occupation time of the particle on a given latticepoint exhibits bimodal U or trimodal W shape, related to the arcsine law. The key pointsare as follows. (a) In a CTRW with finite or infinite mean waiting time, the distribution ofthe number of visits on a lattice point is determined by the probability that a member ofan ensemble of particles in equilibrium occupies the lattice point. (b) The asymmetryparameter of the probability distribution function of occupation times is related to theBoltzmann probability and to the partition function. (c) The ensemble average is givenby Boltzmann–Gibbs statistics for either finite or infinite mean sojourn time,when detailed balance conditions hold. (d) A non-ergodic generalization of theBoltzmann–Gibbs statistical mechanics for systems with infinite mean sojourn time isfound.

ASYMPTOTIC DISTRIBUTIONS AND LARGE DEVIATIONS FOR ITERATED MAPS WITH AN INDIFFERENT FIXED POINT

We consider two-to-one interval maps which are expanding except at an indifferent fixed point. Our purpose is to show that the iteration of the Perron–Frobenius operator displays interesting asymptotic behaviour if we stretch the underlying random variables in a suitable way. As an immediate probabilistic consequence we get a distributional limit theorem, closely related to an arc-sine law, or a large deviation type result, depending on whether the return index is less than or equal to 1.

Distributional limit theorems in infinite ergodic theory

We present a unified approach to the Darling-Kac theorem and the arcsine laws for occupation times and waiting times for ergodic transformations preserving an infinite measure. Our method is based on control of the transfer operator up to the first entrance to a suitable reference set rather than on the full asymptotics of the operator. We illustrate our abstract results by showing that they easily apply to a significant class of infinite measure preserving interval maps. We also show that some of the tools introduced here are useful in the setup of pointwise dual ergodic transformations.

On a generalized arc-sine law for one-dimensional diffusion processes

Laws of the occupation times on a half line are studied for one-dimensional diffusion processes. The asymptotic behavior of the distribution function is determined in terms of the speed measure.

On the quantiles of Brownian motion and their hitting times

The distribution of the α-quantile of a Brownian motion on an interval [0,t] has been obtained motivated by a problem in financial mathematics. In this paper we generalize these results by calculating an explicit expression for the joint density of the α-quantile of a standard Brownian motion, its first and last hitting times and the value of the process at time t. Our results can easily be generalized to a Brownian motion with drift. It is shown that the first and last hitting times follow a transformed arcsine law.

Distributions of Invariant Ensembles from the Classical Orthogonal Polynimials: the Discrete Case

We examine the Charlier, Meixner, Krawtchouk and Hahn discrete orthogonal polynomial ensembles, deeply investigated by K. Johansson, using integration by parts for the underlying Markov operators, differential equations on Laplace transforms and moment equations. As for the matrix ensembles, equilibrium measures are described as limits of empirical spectral distributions. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. Factorial moment identities on mean spectral measures may be used towards small deviation inequalities on the rightmost charges at the rate given by the Tracy-Widom asymptotics.