distributions of time averages for weakly chaotic systems the role of infinite invariant density distributions of time averaged observables are investigated using deterministic maps with n indifferent fixed points and n state continuous time random walk processes associated with them in a weakly chaotic phase namely when separation of trajectories is subexponential maps are characterized by an infinite invariant density we find that the infinite density can be used to calculate the distribution of time averages of integrable observables with a formula recently obtained by rebenshtok and barkai as an example we calculate distributions of the average position of the particle and average occupation fractions our work provides the distributional limit theorem for time averages for a wide class of nonintegrable observables with respect to the infinite invariant density in other words it deals with the situation where the darling kac aaronson theorem does not hold

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

ergodic theory for markov fibred systems and parabolic rational maps a paraboiic rational map of the riemann sphere admits a non atomic h conformal measure on its julia set where h the hausdorff dimension of the julia set and satisfies 1 2 h 2 with respect to this measure the rational map is conservative exact and there is an equivalent a finite invariant measure finiteness of the measure is characterised central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one a theory of markov fibred systems is developed and parabolic rational maps are considered within this framework

aging arcsine law in brownian motion and its generalization classical arcsine law states that the fraction of occupation time on the positive or the negative side in brownian motion does not converge to a constant but converges in distribution to the arcsine distribution here we consider how a preparation of the system affects the arcsine law i e aging of the arcsine law we derive an aging distributional theorem for occupation time statistics in brownian motion where the ratio of time when measurements start to the measurement time plays an important role in determining the shape of the distribution furthermore we show that this result can be generalized as an aging distributional limit theorem in renewal processes

the dynkin lamperti arc sine laws for measure preserving transformations arc sine laws in the sense of renewal theory are proved for return time processes generated by transformations with infinite invariant measure on sets satisfying a type of darling kac condition and an application to real transformations with indifferent fixed points is discussed

gauge potentials and bundles over the 4 torus the construction of principal bundles over a four dimensional torus is considered the class of groups considered is su n zn and for this class the pontrjagin class has even integer values

arcsine laws in stochastic thermodynamics we show that the fraction of time that a thermodynamic current spends above its average value follows the arcsine law a prominent result obtained by levy for brownian motion stochastic currents with long streaks above or below their average are much more likely than those that spend similar fractions of time above and below their average our result is confirmed with experimental data from a brownian carnot engine we also conjecture that two other random times associated with currents obey the arcsine law the time a current reaches its maximum value and the last time a current crosses its average value these results apply to inter alia molecular motors quantum dots and colloidal systems

fluctuations around equilibrium laws in ergodic continuous time random walks we study occupation time statistics in ergodic continuous time random walks under thermal detailed balance conditions the average occupation time is given by the boltzmann gibbs canonical law but close to the nonergodic phase the finite time fluctuations around this mean are large and nontrivial they exhibit dual time scaling and distribution laws the infinite density of large fluctuations complements the levy stable density of bulk fluctuations neither of the two should be interpreted as a stand alone limiting law as each has its own deficiency the infinite density has an infinite norm despite particle conservation while the stable distribution has an infinite variance although occupation times are bounded these unphysical divergences are remedied by consistent use and interpretation of both formulas interestingly while the system s canonical equilibrium laws naturally determine the mean occupation time of the ergodic motion they also control the infinite and levy stable densities of fluctuations the duality of stable and infinite densities is in fact ubiquitous for these dynamics as it concerns the time averages of general physical observables

separation of trajectories and its relation to entropy for intermittent systems with a zero lyapunov exponent i t where is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points the mean of is determined by the infinite invariant density using semianalytical arguments we calculate the infinite invariant density for the pomeaumanneville map and with it we obtain excellent agreement between numerical simulation and theory we show that is equal to krengel s entropy and to the complexity calculated by the lempel ziv compression algorithm this generalized pesin s identity shows that and krengel s entropy are the natural generalizations of usual lyapunov exponent and entropy for these systems

intrinsic randomness of transport coefficient in subdiffusion with static disorder fluctuations in the time averaged mean square displacement for random walks on hypercubic lattices with static disorder are investigated it is analytically shown that the diffusion coefficient becomes a random variable as a manifestation of weak ergodicity breaking for two and higher dimensional systems the distribution function of the diffusion coefficient is found to be the mittag leffler distribution which is the same as for the continuous time random walk whereas for one dimensional systems a different distribution a modified mittag leffler distribution arises we also present a comparison of these two distributions in terms of an ergodicity breaking parameter and show that the modified mittag leffler distribution has a larger deviation from ergodicity some remarks on similarities between these results and observations in biological experiments are presented

generalized arcsine laws for fractional brownian motion the three arcsine laws for brownian motion are a cornerstone of extreme value statistics for a brownian b t starting from the origin and evolving during time t one considers the following three observables i the duration t the process is positive ii the time t mathrm last the process last visits the origin and iii the time t mathrm max when it achieves its maximum or minimum all three observables have the same cumulative probability distribution expressed as an arcsine function thus the name arcsine laws we show how these laws change for fractional brownian motion x t a non markovian gaussian process indexed by the hurst exponent h it generalizes standard brownian motion i e h frac 1 2 we obtain the three probabilities using a perturbative expansion in h ensuremath frac 1 2 while all three probabilities are different this distinction can only be made at second order in our results are confirmed to high precision by extensive numerical simulations

anisotropy and scaling corrections in turbulence we analyze second order turbulent velocity moments both in r and in p space finite size corrections induce dramatic differences between local r and p space scaling exponents as analytically accessible examples we focus on two popular parametrizations the batchelor parametrization for the r space structure function and a common parametrization for the energy spectrum e p is proportional to p 5 3exp p pd the spectral bottleneck energy pileup hidden in the batchelor parametrization results in an extended r space scaling range comparable to experimental ones for the same taylor reynolds number re lambda shear effects are discussed in terms of global apparent scaling correction delta zeta app re lambda to classical scaling which again depend on whether looked at in r or in p space the differences can be traced back to the subtleties of the crossovers in the velocity moments our observations emphasize the need for more experimental information on crossovers between different subranges

number of first passage times as a measurement of information for weakly chaotic systems we consider a general class of maps of the interval having lyapunov subexponential instability xt x0 exp t x0 t where t grows sublinearly as t we outline here a scheme j stat phys 154 988 2014 whereby the choice of a characteristic function automatically defines the map equation and corresponding growth rate t this matching approach is based on the infinite measure property of such systems we show that the average information that is necessary to record without ambiguity a trajectory of the system tends to t suitably extending the kolmogorov sinai entropy and pesin s identity for such systems information behaves like a random variable for random initial conditions its statistics obeying a universal mittag leffler law we show that for individual trajectories information can be accurately inferred by the number of first passage times through a given turbulent phase space cell this enables us to calculate far more efficiently lyapunov exponents for such systems lastly we also show that the usual renewal description of jumps to the turbulent cell usually employed in the literature does not provide the real number of entrances there our results are supported by exhaustive numerical simulations

statistics of occupation times and connection to local properties of nonhomogeneous random walks we consider the statistics of occupation times the number of visits at the origin and the survival probability for a wide class of stochastic processes which can be classified as renewal processes we show that the distribution of these observables can be characterized by a single exponent that is connected to a local property of the probability density function of the process viz the probability of occupying the origin at time t p t we test our results for two different models of lattice random walks with spatially inhomogeneous transition probabilities one of which of non markovian nature and find good agreement with theory we also show that the distributions depend only on the occupation probability of the origin by comparing them for the two systems when p t shows the same long time behavior each observable follows indeed the same distribution

remarks on the yang mills equations in four dimensions cp2 this paper examines properties of solutions to the yang mills equations in four dimensions and in particular on the manifold cp2 two solutions are found one is neither self dual nor anti self dual but is a solution of the full yang mills equations the other is a self dual solution

nonergodicity of a time series obeying levy statistics time averaged autocorrelation functions of a dichotomous random process switching between 1 and 0 and governed by wide power law sojourn time distribution are studied such a process called a levy walk describes dynamical behaviors of many physical systems fluorescence intermittency of semiconductor nanocrystals under continuous laser illumination being one example when the mean sojourn time diverges the process is non ergodic in that case the time average autocorrelation function is not equal to the ensemble averaged autocorrelation function instead it remains random even in the limit of long measurement time several approximations for the distribution of this random autocorrelation function are obtained for different parameter ranges and favorably compared to monte carlo simulations nonergodicity of the power spectrum of the process is briefly discussed and a nonstationary wiener khintchine theorem relating the correlation functions and the power spectrum is presented the considered situation is in full contrast to the usual assumptions of ergodicity and stationarity

random f expansions we consider the asymptotic distribution properties of f expansion digits in particular if x 1 varphi 0 x 1 varphi 1 x cdots etc then frac 1 n sum n 1 k 0 varphi k rightarrow 3 text in measure

distributional ergodicity in stored energy driven levy flights we study a class of random walk the stored energy driven l evy flight sedlf whose jump length is determined by a stored energy during a trapped state the sedlf is a continuous time random walk with jump lengths being coupled with the trapping times it is analytically shown that the ensemble averaged mean square displacements exhibit subdiffusion as well as superdiffusion depending on the coupling parameter we find that time averaged mean square displacements increase linearly with time and the diffusion coefficients are intrinsically random a manifestation of distributional ergodicity the diffusion coefficient shows aging in subdiffusive regime whereas it increases with the measurement time in superdiffusive regime

limit theorems for levy walks in d dimensions rare and bulk fluctuations we consider super diffusive levy walks in dimensions when the duration of a single step i e a ballistic motion performed by a walker is governed by a power law tailed distribution of infinite variance and finite mean we demonstrate that the probability density function pdf of the coordinate of the random walker has two different scaling limits at large times one limit describes the bulk of the pdf it is the d dimensional generalization of the one dimensional levy distribution and is the counterpart of the central limit theorem clt for random walks with finite dispersion in contrast with the one dimensional levy distribution and the clt this distribution does not have a universal shape the pdf reflects anisotropy of the single step statistics however large the time is the other scaling limit the so called infinite density describes the tail of the pdf which determines second dispersion and higher moments of the pdf this limit repeats the angular structure of the pdf of velocity in one step a typical realization of the walk consists of anomalous diffusive motion described by anisotropic d dimensional levy distribution interspersed with long ballistic flights described by infinite density the long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion we illustrate the concept by considering two types of levy walks with isotropic and anisotropic distributions of velocities furthermore we show that for isotropic but otherwise arbitrary velocity distributions the d dimensional process can be reduced to a one dimensional levy walk we briefly discuss the consequences of non universality for the d 1 dimensional fractional diffusion equation in particular the non uniqueness of the fractional laplacian

statistics of the occupation time of renewal processes we present a systematic study of the statistics of the occupation time and related random variables for stochastic processes with independent intervals of time according to the nature of the distribution of time intervals the probability density functions of these random variables have very different scalings in time we analyze successively the cases where this distribution is narrow where it is broad with index 1 and finally where it is broad with index 1 2 the methods introduced in this work provide a basis for the investigation of the statistics of the occupation time of more complex stochastic processes see joint paper by g de smedt c godreche and j m luck 26

lyapunov statistics and mixing rates for intermittent systems we consider here a recent conjecture stating that correlation functions and tail probabilities of finite time lyapunov exponents would have the same power law decay in weakly chaotic systems we demonstrate that this conjecture fails for a generic class of maps of the pomeau manneville type we show further that typically the decay properties of such tail probabilities do not provide significant information on key aspects of weakly chaotic dynamics such as ergodicity and instability regimes our approaches are firmly based on rigorous results particularly the aaronson darling kac theorem and are also confirmed by exhaustive numerical simulations

velocity probability density functions of high reynolds number turbulence abstract this paper deals with the probability density function pdf of velocity differences between two points separated by distance r measurements of pdfs were made for r lying in the inertial range for two different flows in a jet with r 852 and in a wind tunnel with r 2720 these pdfs have a characteristic non gaussian shape with exponential tails following kolmogorov s general ideas of log normality a new model for the pdf is developed which contains two parameters determined by experiments this empirical model agrees with the experimental results that the tails of the pdf deviate from a truly exponential behaviour in particular for small r in addition the model leads to the general scaling law ln e 2 r r 0 differenr from kolmogorov s third hypothesis ln e 2 ln r r 0 restricted to the inertial range only x is x x we develop also a formalism based on an extremum principle which is consistent with both the log normality of e and the above mentioned power law in this formalism can be interpreted as the codimension of dissipative structures and asymptotically varies as 1 ln r

infinite invariant density in a semi markov process with continuous state variables we report on a fundamental role of a non normalized formal steady state i e an infinite invariant density in a semi markov process where the state is determined by the interevent time of successive renewals the state describes certain observables found in models of anomalous diffusion e g the velocity in the generalized levy walk model and the energy of a particle in the trap model in our model the interevent time distribution follows a fat tailed distribution which makes the state value more likely to be zero because long interevent times imply small state values we find two scaling laws describing the density for the state value which accumulates in the vicinity of zero in the long time limit these laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density moreover we provide two distributional limit theorems for time averaged observables in these nonstationary processes we show that the infinite invariant density plays an important role in determining the distribution of time averages

generalized lyapunov exponent as a unified characterization of dynamical instabilities the lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits a positive lyapunov exponent exponential dynamical instability is a manifestation of chaos here we propose the lyapunov pair which is based on the generalized lyapunov exponent as a unified characterization of nonexponential and exponential dynamical instabilities in one dimensional maps chaos is classified into three different types i e superexponential exponential and subexponential chaos using one dimensional maps we demonstrate superexponential and subexponential chaos and quantify the dynamical instabilities by the lyapunov pair in subexponential chaos we show superweak chaos which means that the growth of the difference of nearby orbits is slower than a stretched exponential growth the scaling of the growth is analytically studied by a recently developed theory of a continuous accumulation process which is related to infinite ergodic theory

phase diagram in stored energy driven levy flight phase diagram based on the mean square displacement msd and the distribution of diffusion coefficients of the time averaged msd for the stored energy driven levy flight sedlf is presented in the sedlf a random walker cannot move while storing energy and it jumps by the stored energy the sedlf shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion depending on the coupling parameter between storing time trapping time and stored energy this stochastic process can be investigated analytically with the aid of renewal theory here we consider two different renewal processes i e ordinary renewal process and equilibrium renewal process when the mean trapping time does not diverge we analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping time distribution in particular we find that distributional behavior of time averaged msd intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge

log similarity for turbulent flows abstract we critically examine a recently proposed representation for turbulent energy spectra called here log similarity which is predicted both by the multifractal model and the variational approach of the small scale intermittency within this representation spectra are homothetic in the log log plot the homothety factor depending on the taylor scale based reynolds number r experimental results on energy spectra and velocity structure functions of order 2 3 and 6 measured in different flows at several r show that this factor is r a measurable exponent which characterises intermittency in the inertial range experimentally displays the existence of a transition in the neighbourhood of r 400 and for r tending to infinity behaves as 1 in r r

fractional feynman kac equation for weak ergodicity breaking the continuous time random walk ctrw is a model of anomalous subdiffusion in which particles are immobilized for random times between successive jumps a power law distribution of the waiting times 1 leads to subdiffusion x 2 t for 0 1 in closed systems the long stagnation periods cause time averages to divert from the corresponding ensemble averages which is a manifestation of weak ergodicity breaking the time average of a general observable u t 1 t 0 t u x d is a functional of the path and is described by the well known feynman kac equation if the motion is brownian here we derive forward and backward fractional feynman kac equations for functionals of ctrw in a binding potential we use our equations to study two specific time averages the fraction of time spent by a particle in half box and the time average of the particle s position in a harmonic field in both cases we obtain the probability density function of the time averages for t and the first two moments our results show that both the occupation fraction and the time averaged position are random variables even for long times except for 1 when they are identical to their ensemble averages using our fractional feynman kac equation we also study the dynamics leading to weak ergodicity breaking namely the convergence of the fluctuations to their asymptotic values

galton watson iterated function systems iterated function systems ifs are interesting parametric models for generating fractal sets and functions the general idea is to compress deform and translate a given set or function with a collection of operators and to iterate the procedure under weak assumptions ifs possess a unique fixed point which is in general fractal ifs were introduced in a deterministic context then were generalized to the random setting on abstract spaces in the early 1980 s their adaptation to random signals was carried out by hutchinson and r schendorff 9 by considering random operators this study extends their model with not only random operators but also a random underlying construction tree we show that the corresponding ifs converges under certain hypothesis to a unique fractal fixed point properties of the fixed point are also described

some twisted self dual solutions for the yang mills equations on a hypertorus thesu n yang mills equations are considered in a four dimensional euclidean box with periodic boundary conditions hypertorus gauge invariant twists can be introduced in these boundary conditions to be labeled with integersn n defined modulon the pontryagin number in this space is often fractional whenever this number is zero there are solutions to the equationsg 0 hereg is the covariant curl when this number is not zero we find a set of solutions to the equations g mu nu tilde g mu nu provided that the periodsa of the box satisfy certain relations

pseudononstationarity in the scaling exponents of finite interval time series the accurate estimation of scaling exponents is central in the observational study of scale invariant phenomena natural systems unavoidably provide observations over restricted intervals consequently a stationary stochastic process time series can yield anomalous time variation in the scaling exponents suggestive of nonstationarity the variance in the estimates of scaling exponents computed from an interval of n observations is known for finite variance processes to vary as ensuremath sim 1 n as n ensuremath rightarrow ensuremath infty for certain statistical estimators however the convergence to this behavior will depend on the details of the process and may be slow we study the variation in the scaling of second order moments of the time series increments with n for a variety of synthetic and real world time series and we find that in particular for heavy tailed processes for realizable n one is far from this ensuremath sim 1 n limiting behavior we propose a semiempirical estimate for the minimum n needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some real world time series

distributional behavior of time averages of non l 1 observables in one dimensional intermittent maps with infinite invariant measures in infinite ergodic theory two distributional limit theorems are well known one is characterized by the mittag leffler distribution for time averages of l 1 m functions i e integrable functions with respect to an infinite invariant measure the other is characterized by the generalized arc sine distribution for time averages of non l 1 m functions here we provide another distributional behavior of time averages of non l 1 m functions in one dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure observation functions considered here are non l 1 m functions which vanish at the indifferent fixed point we call this class of observation functions weak non l 1 m function our main result represents a first step toward a third distributional limit theorem i e a distributional limit theorem for this class of observables in infinite ergodic theory to prove our proposition we propose a stochastic process induced by a renewal process to mimic a birkoff sum of a weak non l 1 m function in the one dimensional intermittent maps

singularity spectrum of fractal signals from wavelet analysis exact results the multifractal formalism for singular measures is revisited using the wavelet transform for bernoulli invariant measures of some expanding markov maps the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima the generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures it is demonstrated that the hausdorff dimensiond h of the set of singularities of holder exponenth can be directly determined from the wavelet transform modulus maxima the singularity spectrum so obtained is shown to be not disturbed by the presence in the signal of a superimposed polynomial behavior of ordern provided one uses an analyzing wavelet that possesses at leastn n vanishing moments however it is shown that ac behavior generally induces a phase transition in thed h singularity spectrum that somewhat masks the weakest singularities this phase transition actually depends on the numbern of vanishing moments of the analyzing wavelet its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal these theoretical results are illustrated with numerical examples they are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional brownian motions and turbulent velocity signals

edges transitions and criticality in this article various notions of edges encountered in digital image processing are reviewed in terms of compact representation or completion we show that critical exponents defined in statistical physics lead to a much more coherent definition of edges consistent across the scales in acquisitions of natural phenomena such as high resolution natural images or turbulent acquisitions edges belong to the multiscale hierarchy of an underlying dynamics they are understood from a statistical perspective well adapted to fit the case of natural images numerical computation methods for the evaluation of critical exponents in the non ergodic case are recalled which apply for the vast majority of natural images we study the framework of reconstructible systems in a microcanonical formulation show how it redefines edge completion and how it can be used to evaluate and assess quantitatively the adequation of edges as candidates for compact representations we study with particular attention the case of turbulent data in which edges in the classical sense are particularly challenged tests are conducted and evaluated on a standard database for natural images we test the newly introduced compact representation as an ideal candidate for evaluating turbulent cascading properties of complex images and we show better reconstruction performance than the classical tested methods highlightsclassical edge detection algorithms are reviewed in terms of completion a new edge detection scheme based on multiscale hierarchical classification assessing the consistency of edges across scales evaluating edge performance through the framework of reconstructible systems comparing the performance with classical edge operators

treatment of kurtosis in financial markets abstract since mandelbrot 1963 2 highlighted the fact that data on the yield of financial assets exhibit leptokurtosis different distributions have been presented as alternatives to the normal distribution so far little consideration has been given to the capacity that these distributions have to recover the kurtosis of the sample data our work aims to present distributions which given the broad range of their kurtosis have the capacity to perform adjustment on many occasions where other distributions fail while also being capable of recovering the peakedness of the empirical data another key characteristic of these distributions is that they are defined within a bounded domain in the same way as the sample data an empirical application of these distributions is presented within the financial field by using daily returns

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

testing for the source of multifractality in water level records abstract the multifractality in hydrologic data has been studied in many literatures but few literatures focus on its source which is helpful for understanding the underlying mechanisms that generate hydrologic data we propose a hypothesis testing procedure for the source of multifractality in water level records based on the empirical distributions of generalized hurst exponent estimated from a set of shuffled or surrogate series the proposed hypothesis testing procedure can show more details about the source of multifractality than previous methods with some statistics especially about the effects of large and small fluctuations on multifractality the generalized hurst exponents are estimated via multifractal detrended fluctuation analysis the data set contains about two million high frequency water level records of a northern china river at its 10 observation stations the testing results show that the multifractality in water levels is mainly caused by nonlinear correlations in small fluctuations and linear correlations in small and large fluctuations and is also related to the probability distribution of small fluctuations this conclusion is validated via some constructed semi surrogate series and values of statistics for source testing

data driven detrending of nonstationary fractal time series with echo state networks abstract in this paper we propose a novel data driven approach for removing trends detrending from nonstationary fractal and multifractal time series we consider real valued time series relative to measurements of an underlying dynamical system that evolves through time we assume that such a dynamical process is predictable to a certain degree by means of a class of recurrent networks called echo state network esn which are capable to model a generic dynamical process in order to isolate the superimposed multi fractal component of interest we define a data driven filter by leveraging on the esn prediction capability to identify the trend component of a given input time series specifically the estimated trend is removed from the original time series and the residual signal is analyzed with the multifractal detrended fluctuation analysis procedure to verify the correctness of the detrending procedure in order to demonstrate the effectiveness of the proposed technique we consider several synthetic time series consisting of different types of trends and fractal noise components with known characteristics we also process a real world dataset the sunspot time series which is well known for its multifractal features and has recently gained attention in the complex systems field results demonstrate the validity and generality of the proposed detrending method based on esns

wavelet based estimation of 1 f type signal parameters confidence intervals using the bootstrap we propose to construct confidence intervals of parameters of 1 f type signals using a nonparametric wavelet based bootstrap method bootstrap based confidence intervals of maximum likelihood parameter estimates are compared to the confidence intervals derived from the cramer rao lower bound crlb for moderately large data sample sizes the bootstrap approach achieves the nominal coverage and may perform better than the crlb based parametric approach

renewal theory with fat tailed distributed sojourn times typical versus rare renewal processes with heavy tailed power law distributed sojourn times are commonly encountered in physical modeling and so typical fluctuations of observables of interest have been investigated in detail to describe rare events the rate function approach from large deviation theory does not hold and new tools must be considered here we investigate the large deviations of the number of renewals the forward and backward recurrence times the occupation time and the time interval straddling the observation time we show how non normalized densities describe these rare fluctuations and how moments of certain observables are obtained from these limiting laws numerical simulations illustrate our results showing the deviations from arcsine dynkin darling kac l evy and lamperti laws

how much information does dependence between wavelet coefficients contain abstractthis article is motivated by several articles that propose statistical inference where the independence of wavelet coefficients for both short as well as long range dependent time series is assumed we focus on the sample variance and investigate the influence of the dependence between wavelet coefficients and this statistic to this end we derive asymptotic distributional properties of the sample variance for a time series that is synthesized ignoring some or all dependence between wavelet coefficients we show that the second order properties differ from the those of the true time series whose wavelet coefficients have the same marginal distribution except in the independent gaussian case this holds true even if the dependency is correct within each level and only the dependence between levels is ignored in the case of sample autocovariances and sample autocorrelations at lag one we indicate that first order properties are erroneous in a second step we investigate several nonparametric b

self similar anisotropic texture analysis the hyperbolic wavelet transform contribution textures in images can often be well modeled using self similar processes while they may simultaneously display anisotropy the present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of gaussian anisotropic selfsimilar processes it will be first shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard 2d discrete wavelet transform with the hyperbolic wavelet transform which permits the use of different dilation factors along the horizontal and vertical axes defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized this discrepancy defines a rotation angle second we show that this rotation angle can be jointly estimated third a nonparametric bootstrap based procedure is described which provides confidence intervals in addition to the estimates themselves and enables us to construct an isotropy test procedure which can be applied to a single texture image fourth the robustness and versatility of the proposed analysis are illustrated by being applied to a large variety of different isotropic and anisotropic self similar fields as an illustration we show that a true anisotropy built in self similarity can be disentangled from an isotropic self similarity to which an anisotropic trend has been superimposed

unified multifractal description of velocity increments statistics in turbulence intermittency and skewness abstract the phenomenology of velocity statistics in turbulent flows up to now relates to different models dealing with either signed or unsigned longitudinal velocity increments with either inertial or dissipative fluctuations in this paper we are concerned with the complete probability density function pdf of signed longitudinal increments at all scales first we focus on the symmetric part of the pdfs taking into account the observed departure from scale invariance induced by dissipation effects the analysis is then extended to the asymmetric part of the pdfs with the specific goal to predict the skewness of the velocity derivatives it opens the route to the complete description of all measurable quantities for any reynolds number and various experimental conditions this description is based on a single universal parameter function d h and a universal constant r

mixing rates for intermittent maps of high exponent we obtain higher order theory for the long term behavior of the transfer operator associated with the unit interval map f x x 1 2 alpha x alpha if 0 1 which corresponds to the infinite measure preserving case higher order theory for alpha ge 2 is more challenging and requires new techniques along the way we provide higher order theory for scalar and operator renewal sequences with infinite measure and regular variation although the present work considers the unit interval map mentioned above as a toy model our interest focuses on finding sufficient conditions under which the asymptotic behavior of the transfer operator associated to dynamical systems preserving an infinite measure is almost like the asymptotic behavior of scalar renewal sequences associated to null recurrent markov chains characterized by regular variation

multifractality tests using bootstrapped wavelet leaders multifractal analysis which mostly consists of measuring scaling exponents is becoming a standard technique available in most empirical data analysis toolboxes making use of the most recent theoretical results it is based here on the estimation of the cumulants of the log of the wavelet leaders an elaboration on the wavelet coefficients these log cumulants theoretically enable discrimination between mono and multifractal processes as well as between simple log normal multifractal models and more advanced ones the goal of the present contribution is to design nonparametric bootstrap hypothesis tests aiming at testing the nature of the multifractal properties of stochastic processes and empirical data bootstrap issues together with six declinations of test designs are analyzed their statistical performance significances powers and p values are assessed and compared by means of monte carlo simulations performed on synthetic stochastic processes whose multifractal properties and log cumulants are known theoretically a priori we demonstrate that the joint use of wavelet leaders log cumulants and bootstrap procedures enable us to obtain a powerful tool for testing the multifractal properties of data this tool is practically effective and can be applied to a single observation of data with finite length

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

multifractal cross correlation effects in two variable time series of complex network vertex observables we investigate the scaling of the cross correlations calculated for two variable time series containing vertex properties in the context of complex networks time series of such observables are obtained by means of stationary unbiased random walks we consider three vertex properties that provide respectively short medium and long range information regarding the topological role of vertices in a given network in order to reveal the relation between these quantities we applied the multifractal cross correlation analysis technique which provides information about the nonlinear effects in coupling of time series we show that the considered network models are characterized by unique multifractal properties of the cross correlation in particular it is possible to distinguish between erdos renyi barabasi albert and watts strogatz networks on the basis of fractal cross correlation moreover the analysis of protein contact networks reveals characteristics shared with both scale free and small world models

an improved multifractal formalism and self similar measures abstract to characterize the geometry of a measure its generalized dimensions dq have been introduced recently the mathematically precise definition given by falconer fractal geometry 1990 turns out to be unsatisfactory for reasons of convergence as well as of undesired sensitivity to the particular choice of coordinates in the negative q range a new definition is introduced which is also based on box counting but which carries relevant information about for negative q also in particular the legendre connection between generalized dimensions and the multifractal spectrum is established using large deviations and rigorous proofs are provided for the implicit formula giving the generalized dimensions of selfsimilar measures which was until now known only for positive q

operator renewal theory and mixing rates for dynamical systems with infinite measure we develop a theory of operator renewal sequences in the context of innite ergodic theory for large classes of dynamical systems preserving an innite measure we determine the asymptotic behaviour of iterates l n of the transfer operator this was previously an intractable problem examples of systems covered by our results include i parabolic rational maps of the complex plane and ii not necessarily markovian nonuniformly expanding interval maps with indierent xed points in addition we give a particularly simple proof of pointwise dual ergodicity asymptotic behaviour of p n 1 l j for the class of systems under consideration in certain situations including pomeau manneville intermittency maps we obtain higher order expansions for l n and rates of mixing also we obtain error estimates in the associated dynkin lamperti arcsine laws

aging generates regular motions in weakly chaotic systems using intermittent maps with infinite invariant measures we investigate the universality of time averaged observables under aging conditions according to aaronson s darling kac theorem in non aged dynamical systems with infinite invariant measures the distribution of the normalized time averages of integrable functions converges to the mittag leffler distribution this well known theorem holds when the start of observations coincides with the start of the dynamical processes introducing a concept of the aging limit where the aging time t a and the total measurement time t goes to infinity while the aging ratio t a t is kept constant we obtain a novel distributional limit theorem of time averaged observables integrable with respect to the infinite invariant density applying the theorem to the lyapunov exponent in intermittent maps we find that regular motions and a weakly chaotic behavior coexist in the aging limit this mixed type of dynamics is controlled by the aging ratio and hence is very different from the usual scenario of regular and chaotic motions in hamiltonian systems the probability of finding regular motions in non aged processes is zero while in the aging regime it is finite and it increases when system ages

distributional response to biases in deterministic superdiffusion we report on a novel response to biases in deterministic superdiffusion for its reduced map we show using infinite ergodic theory that the time averaged velocity tav is intrinsically random and its distribution obeys the generalized arcsine distribution a distributional limit theorem indicates that the tav response to a bias appears in the distribution which is an example of what we term a distributional response induced by a bias although this response in single trajectories is intrinsically random the ensemble averaged tav response is linear

two guidelines for bootstrap hypothesis testing two guidelines for nonparametric bootstrap hypothesis testing are highlighted the first recommends that resampling be done in a way that reflects the null hypothesis even when the true hypothesis is distant from the null the second guideline argues that bootstrap hypothesis tests should employ methods that are already recognized as having good features in the closely related problem of confidence interval construction violation of the first guideline can seriously reduce the power of a test sometimes this reduction is spectacular since it is most serious when the null hypothesis is grossly in error the second guideline is of some importance when the conclusion of a test is equivocal it has no direct bearing on power but improves the level accuracy of a test

anomalous diffusion in a quenched trap model on fractal lattices models with mixed origins of anomalous subdiffusion have been considered important for understanding transport in biological systems here one such mixed model the quenched trap model qtm on fractal lattices is investigated it is shown that both ensemble and time averaged mean square displacements msds show subdiffusion with different scaling exponents i e this system shows weak ergodicity breaking moreover time averaged msd exhibits aging and converges to a random variable following the modified mittag leffler distribution it is also shown that the qtm on a fractal lattice cannot be reduced to the continuous time random walks if the spectral dimension of the fractal lattice is less than 2

empirical distributions of chinese stock returns at different microscopic timescales abstract we study the distributions of event time returns and clock time returns at different microscopic timescales using ultra high frequency data extracted from the limit order books of 23 stocks traded in the chinese stock market in 2003 we find that the returns at the one trade timescale obey the inverse cubic law for larger timescales 2 32 trades and 1 5 min the returns follow the student distribution with power law tails with the decrease in timescale the tail becomes fatter which is consistent with the variational theory in turbulence