asymptotic normality of goodness of fit statistics for sparse product multinomials summary the asymptotic distribution of the pearson x2 and likelihood ratio g2 goodness of fit statistics is derived for k independent non identically distributed multinomials as k approaches infinity with expected cell frequencies bounded the application is to large datasets of counts with many zero cells conditions are given under which the limiting distributions are normal for originating probabilities i known and ii dependent on a common finite dimensional parameter a logistic regression model is given which satisfies ii finally skewness is discussed as a measure of closeness of moderate upper percentage points of x2 to normal percentiles

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

divergence statistics for testing uniform association in cross classifications in this paper we consider the problem of testing uniform association in cross classifications having ordered categories taking as test statistic one in the family proposed by conde and salicru j conde m salicru uniform association in contingency tables associated to csiszar divergence statistics and probability letters 37 1998 149 154 we consider two approximations to the null distribution of the test statistics in this family an estimation of the asymptotic null distribution and a bootstrap estimator we prove that both approximations are asymptotically equivalent to study their finite sample performance we carried out two simulation experiments whose results are presented from the simulations it can be concluded that the bootstrap estimator behaves much better than the estimated asymptotic null distribution

random walks on graphs ideas techniques and results random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology in recent years relevant mathematical results have been obtained in this field and new ideas have been introduced which can be fruitfully extended to different areas and disciplines here we aim at giving a brief but comprehensive perspective of these progresses with a particular emphasis on physical aspects

random walk in chemical space of cantor dust as a paradigm of superdiffusion we point out that the chemical space of a totally disconnected cantor dust k n ensuremath subset e n is a compact metric space c n with the spectral dimension d s d ensuremath ell ngd where d and d ensuremath ell n are the fractal and chemical dimensions of k n respectively hence we can define a random walk in the chemical space as a markovian gaussian process the mapping of a random walk in c n into k n ensuremath subset e n defines the quenched l evy flight on the cantor dust with a single step duration independent of the step length the equations describing the superdiffusion and diffusion reaction front propagation ruled by the local quenched l evy flight on k n ensuremath subset e n are derived the use of these equations to model superdiffusive phenomena observed in some physical systems in which propagators decay faster than algebraically is discussed

multidimensional solutions of space fractional diffusion equations green s functions and propagators for the multidimensional anisotropic spacetime fractional diffusion equation are obtained in integral form uniqueness of the solutions is studied in the framework

hierarchical spatio temporal coupling in fractional wanderings i continuous time weierstrass flights abstract the one dimensional continuous time weierstrass flights ctwf model is considered in the framework of the nonseparable continuous time random walks formalism ctrw a novel spatio temporal coupling is introduced by assuming that in each scale the probability density for the flight and for waiting are joined hence we treat the spatio temporal relations in terms of the self similar structure of the weierstrass process this stochastic structure is characterized by the spatial fractional dimension 1 representing the flights and the temporal one 1 representing the waiting time was assumed here as the only independent truncation range in this work we study the asymptotic properties of the ctwf model for example by applying the method of steepest descents we obtained the particle propagator in the approximate scaling form p x t t pt 2 f where the scalingfunction f pt exp const pt while the scaling variable x t pt 2 is large the principal result of our analysis is that the exponents and depend on more fundamental ones and what leads to a novel scaling as a result of competition between exponents and an enhanced dispersive or normal diffusion was recognized in distinction from the prediction of the separable ctrw model where the enhanced diffusion is lost and the dispersive one is strongly limited it should be noted that we compare here partially thermalized versions of both approaches where some initial fluctuations were also included in agreement with the spirit of the theory of the renewal processes having the propogator we calculated for example the particle mean square displacement and found its novel asymptotic scaling with time for enhanced diffusion given by t1 2 1 in distinction from its diverging for

scale invariant motion in intermittent chaotic systems we investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous time random walk ctrw approach two different ctrw models are considered i particles jump between sites turning points or ii particles move at a constant velocity between sites and choose a new direction at random for both models we study the mean squared displacement mathit r 2 t and the propagator p r t the probability to be at location r at time t having started at the origin at t 0 iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the ctrw framework and scaling arguments for the case of dispersive motion we discuss the problem of the stationary state and point out its relevance

shape analysis via oriented distance functions the object of this paper is twofold we first present constructions which induce topologies on subsets of a fixed domain or hold all d in rn by using set parameterized functions in an appropriate function space second we study the role of the family of oriented distance functions also known as algebraic or signed distance functions in the analysis of shape optimization problems they play an important role in the introduction of topologies which retain the classical geometric properties associated with sets convexity exterior normals mean curvature ck boundaries etc

levy walks in random scattering and growth of waves random spatial wave scattering and stochastic wave growth are studied where one or both of the random processes can be described by a levy walk this analysis extends previous work on randomly growing and scattering waves where both the random processes are modeled by gaussian diffusive statistics both random spatial scattering and stochastic wave growth modeled by levy walks are studied separately together and in combination with gaussian processes transmission coefficients lasing thresholds and energy densities in the medium are obtained for the different permutations

the polya information divergence extensions of sanov s theorem and the conditional limit theorem colt are established for a multicolor polya eggenberger pe urn sampling scheme giving the polya information divergence and a polya extension to the maximum relative entropy maxent method polya maxent includes the standard maxent as well as its variants used in bose einstein fermi dirac and intermediate acharya swamy statistics as special cases in the pe setting standard maxent is in general asymptotically inconsistent

saddlepoint approximations for distributions of quadratic forms in normal variables reference stat article 1999 001doi 10 1093 biomet 86 4 929view record in web of science record created on 2006 04 21 modified on 2017 05 12

levy walks and scaling in quenched disordered media we study levy walks in quenched disordered one dimensional media with scatterers spaced according to a long tailed distribution by analyzing the scaling relations for the random walk probability and for the resistivity in the equivalent electric problem we obtain the asymptotic behavior of the mean square displacement as a function of the exponent characterizing the scatterers distribution we demonstrate that in quenched media different average procedures can display different asymptotic behavior in particular we estimate the moments of the displacement averaged over processes starting from scattering sites our results are compared with numerical simulations with excellent agreement

the analysis of categorical data from complex sample surveys chi squared tests for goodness of fit and independence in two way tables abstract the effect of stratification and clustering on the asymptotic distributions of standard pearson chi squared test statistics for goodness of fit simple hypothesis and independence in a two way contingency table denoted as x 2 and xi 2 respectively is investigated it is shown that both x 2 and xi 2 are asymptotically distributed as weighted sums of independent 1 2 random variables the weights are then related to the familiar design effects deffs used by survey samplers a simple correction to x 2 which requires only the knowledge of variance estimates or deffs for individual cells in the goodness of fit problem is proposed and empirical results on the performance of corrected x 2 provided empirical work on xi 2 indicated that the distortion of nominal significance level is substantially smaller with xi 2 than with x 2 some results under simple models for clustering are also given

one dimensional stochastic levy lorentz gas we introduce a l evy lorentz gas in which a light particle is scattered by static point scatterers arranged on a line we investigate the case where the intervals between scatterers ensuremath xi i are independent random variables identically distributed according to the probability density function ensuremath mu ensuremath xi ensuremath sim ensuremath xi ensuremath 1 ensuremath gamma we show that under certain conditions the mean square displacement of the particle obeys x 2 t g mathrm ct 3 ensuremath ensuremath gamma for 1l ensuremath gamma l2 this behavior is compatible with a renewal l evy walk scheme we discuss the importance of rare events in the proper characterization of the diffusion process

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

exact invariant measures how the strength of measure settles the intensity of chaos the aim of this paper is to show how to extract dynamical behavior and ergodic properties from deterministic chaos with the assistance of exact invariant measures on the one hand we provide an approach to deal with the inverse problem of finding nonlinear interval maps from a given invariant measure then we show how to identify ergodic properties by means of transitions along the phase space via exact measures on the other hand we discuss quantitatively how infinite measures imply maps having subexponential lyapunov instability weakly chaotic as opposed to finite measure ergodic maps which are fully chaotic in addition we provide general solutions of maps for which infinite invariant measures are exactly known throughout the interval a demand from this field finally we give a simple proof that infinite measure implies universal mittag leffler statistics of observables rather than narrow distributions typically observed in finite measure ergodic maps

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

image processing tools for better incorporation of 4d seismic data into reservoir models reservoir engineers have to predict the behavior of a hydrocarbon reservoir by building a simulation model which can best reproduce the data collected in the field these data fall into two types static data which are invariable in time and dynamic data which evolve according to fluid motions in the reservoir in this paper we focus on the integration of dynamic data related to four dimensional 4d inverted seismic data such seismic data constitute an invaluable source of information on fluid displacement and geology over extensive areas of the reservoir however incorporating them in the reservoir model through a matching process is a challenging task classical formulations of the objective function which computes the misfit between observed data and responses computed by the reservoir model are not adapted to 4d inverted seismic data for example a least square based mismatch is not representative of the visual difference between two seismic images in this paper we define a new formulation of the objective function based on simplification of seismic data in order to extract relevant information this simplification involves filtering and segmentation techniques as well as image comparison methods rooted in image analysis more precisely we focus on the non local means algorithm for filtering on the level set framework for segmentation and on the local modified hausdorff distance for image comparison we investigate the efficiency of such techniques in the context of seismic data and illustrate their potential on a synthetic history matching reservoir example

strongly and weakly self similar diffusion many dispersive processes have moments of displacements with large t behavior x p t p the study of p as a function of p provides a more complete characterization of the process than does the single number 2 also at long times the core of the concentration relaxes to a self similar profile while the large x tails consisting of particles which have experienced exceptional displacements are not self similar depending on the particular process the effect of the tails can be negligible and then p is a linear function of p strong self similarity but if the tails are important then p is a non trivial function of p weak self similarity in the weakly self similar case the low moments are determined by the self similar core while the high moments are determined by the non self similar tails the popular exponent 2 may be determined by either the core or the tails as representatives of a large class of dispersive processes for which p is a piecewise linear function of p we study two systems a stochastic model the generalized telegraph model and a deterministic area preserving map the

levy processes on a generalized fractal comb comb geometry constituted of a backbone and fingers is one of the most simple paradigm of a two dimensional structure where anomalous diffusion can be realized in the framework of markov processes however the intrinsic properties of the structure can destroy this markovian transport these effects can be described by the memory and spatial kernels in particular the fractal structure of the fingers which is controlled by the spatial kernel in both the real and the fourier spaces leads to the levy processes levy flights and superdiffusion this generalization of the fractional diffusion is described by the riesz space fractional derivative in the framework of this generalized fractal comb model levy processes are considered and exact solutions for the probability distribution functions are obtained in terms of the fox h function for a variety of the memory kernels and the rate of the superdiffusive spreading is studied by calculating the fractional moments for a special form of the memory kernels we also observed a competition between long rests and long jumps finally we considered the fractal structure of the fingers controlled by a weierstrass function which leads to the power law kernel in the fourier space this is a special case when the second moment exists for superdiffusion in this competition between long rests and long jumps

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

the discrete degree of metric spaces and the lipschitz continuity of distance transforms abstract a quantitative characteristic named the discrete degree of a metric space is introduced in terms of such a characteristic relations between the lipschitz continuity of distance transforms and the properties of the metrics on spaces are discussed

random walks in a one dimensional levy random environment we consider a generalization of a one dimensional stochastic process known in the physical literature as levy lorentz gas the process describes the motion of a particle on the real line in the presence of a random array of marked points whose nearest neighbor distances are i i d and long tailed with finite mean but possibly infinite variance the motion is a continuous time constant speed interpolation of a symmetric random walk on the marked points we first study the quenched random walk on the point process proving the clt and the convergence of all the accordingly rescaled moments then we derive the quenched and annealed clts for the continuous time process

renewal modulation and superstatistics in times series we consider two different approaches to which we refer to as renewal and modulation to generate time series with a nonexponential distribution of waiting times we show that different time series with the same waiting time distribution are not necessarily statistically equivalent and might generate different physical properties renewal generates aging and anomalous scaling while modulation yields no significant aging and either ordinary or anomalous diffusion according to the dynamic prescription adopted we show in fact that the physical realization of modulation generates two classes of events the events of the first class are determined by the persistent use of the same exponential time scale for an extended lapse of time and consequently are numerous the events of the second class are identified with the abrupt changes from one to another exponential prescription and consequently are rare the events of the second class although rare determine the scaling of the diffusion process and for this reason we term them as crucial events according to the prescription adopted to produce modulation the distribution density of the time distances between two consecutive crucial events might have or not a diverging second moment in the former case the resulting diffusion process although going through a transition regime very extended in time will eventually become anomalous in conclusion modulation rather than ruling out the action of renewal events produces crucial events hidden by clouds of exponential events thereby setting the challenge for their identification

simulation study of the tests of uniform association based on the power divergence in this paper a simulation study is presented to analyze the behavior of the family of test statistics proposed by conde and salicru j conde m salicru uniform association in contingency tables associated to csiszar divergence statistics and probability letters 37 1998 149 154 using the phi divergence measures that include as special case the power divergence n cressie t r c read multinomial goodness of fit tests journal of the royal statistic society series b 46 1984 440 464 for the analysis of uniform association between two classification processes based on the local odd ratios for the above test statistics the significance level and its power are evaluated for different sample sizes when we consider a 3 x 2 contingency table

what is the alternative to the alexander orbach relation the alexander orbach ao relation d w 2d f d s connecting the fractal dimension of a random walk s rw trajectory d w or the exponent of anomalous diffusion 2 d w on a fractal structure with the fractal and spectral dimension of the structure itself plays a key role in discussion of dynamical properties of complex systems including living cells and single biomolecules this relation however does not hold universally and breaks down for some structures like diffusion limited aggregates and eden trees we show that the alternative to the ao relation is the explicit dependence of the coefficient of the anomalous diffusion on the system s size i e the absence of its thermodynamical limit the prerequisite for its breakdown is the dependence of the local structure of possible steps of the rw on the system s size the discussion is illustrated by the examples of diffusion on a koch curve ao conform and on a cantor dust violating ao relation

solutions for a fractional nonlinear diffusion equation spatial time dependent diffusion coefficient and external forces we analyze a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation we start our investigation by considering the linear case and the nonlinear case afterward the linear case is discussed taking fractional time and spatial derivatives into account in a unified approach we also discuss the modifications that emerge by employing simple drifts and the diffusion coefficient given by d x t d t x for the nonlinear case we study scaling behavior of the time in connection with the asymptotic behavior for the solution of the nonlinear fractional diffusion equation

distance transforms for real valued functions a set in a metric space gives rise to its distance function that associates with every point its distance to the nearest point in the set this function is called the distance transform of the original set in the same vein given a real valued function f we consider the expected distances from any point to a level set of f taken at a random height this produces another function called a distance transform of f such transforms are called grey scale distance transforms to signpost their differences from the binary case when sets or their indicators give rise to conventional distance functions basic properties of the introduced grey scale distance transform are discussed the most important issue is the uniqueness problem whether two different functions may share the same distance transform we answer this problem in a generality completely sufficient for all practical applications in imaging sciences the full scale problem remains open

holey random walks optics of heterogeneous turbid composites we present a probabilistic theory of random walks in turbid media with nonscattering regions it is shown that important characteristics such as diffusion constants average step lengths crossing statistics and void spacings can be analytically predicted the theory is validated using monte carlo simulations of light transport in heterogeneous systems in the form of random sphere packings and good agreement is found the role of step correlations is discussed and differences between unbounded and bounded systems are investigated our results are relevant to the optics of heterogeneous systems in general and represent an important step forward in the understanding of media with strong fractal heterogeneity in particular

extended rc association models allowing for order restrictions and marginal modeling in the context of two way contingency tables the rc model may be seen as a way of modeling the set of log odds ratios for adjacent 2 2 subtables through row and column scores this article presents an extended class of rc models that allows simultaneous modeling of the marginal distributions and the association between two categorical variables these components are parameterized with logits and log odds ratios that may be of different types e g global or continuation to suit the nature of the data and the type of dependence of interest an algorithm for maximum likelihood estimation under equality and inequality constraints is presented the asymptotic distribution of the likelihood ratio statistic to test some interesting hypotheses is derived and extensions to multiway tables are outlined three examples are presented to highlight the features of the new approach

levy walk approach to anomalous diffusion abstract the transport properties of levy walks are discussed in the framework of continuous time random walks ctrw with coupled memories this type of walks may lead to anomalous diffusion where the mean squared displacement r2 t t with 1 we focus on the enhanced diffusion limit 1 in one dimension and present our results on r2 t the mean number of distinct sites visited s t and p r t the probability of being at position r at time t

single big jump principle in physical modeling the big jump principle is a well established mathematical result for sums of independent and identically distributed random variables extracted from a fat tailed distribution it states that the tail of the distribution of the sum is the same as the distribution of the largest summand in practice it means that when in a stochastic process the relevant quantity is a sum of variables the mechanism leading to rare events is peculiar instead of being caused by a set of many small deviations all in the same direction one jump the biggest of the lot provides the main contribution to the rare large fluctuation we reformulate and elevate the big jump principle beyond its current status to allow it to deal with correlations finite cutoffs continuous paths memory and quenched disorder doing so we are able to predict rare events using the extended big jump principle in l evy walks in a model of laser cooling in a scattering process on a heterogeneous structure and in a class of l evy walks with memory we argue that the generalized big jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy tailed distributions ranging from contamination spreading to active transport in the cell

analytic expression for the mean time to absorption for a random walker on the sierpinski gasket the exact analytic expression for the mean time to absorption or mean walk length for a particle performing a random walk on a finite sierpinski gasket with a trap at one vertex is found to be t n 3 n 5 n 1 4 5 n ensuremath 3 n 3 n 1 1 where n denotes the generation index of the gasket and the mean is over a set of starting points of the walk distributed uniformly over all the other sites of the gasket in terms of the number n n of sites on the gasket and the spectral dimension stackrel ifmmode tilde else fi d of the gasket the precise asymptotic behavior for large n n is t n ensuremath rightarrow 1 3 2n n 2 stackrel ifmmode tilde else fi d ensuremath sim n 1 464 this serves as a partial check on our result as it is a intermediate between the known results t ensuremath sim n 2 d 1 and t ensuremath sim n mathrm ln n d 2 for random walks on d dimensional euclidean lattices and b consistent with the known result for the asymptotic behavior of the mean number of distinct sites visited in a random walk on a fractal lattice

generalized linear boltzmann equation describing non classical particle transport and related asymptotic solutions for small mean free paths abstract in classical kinetic models a particle free path distribution is exponential but this is more likely to be an exception than a rule in this paper we derive a generalized linear boltzmann equation glbe for a general free path distribution in the framework of alt s model in the case that the free path distribution has at least first and second finite moments we construct an asymptotic solution to the initial value problem for the glbe for small mean free paths in the special case of the one speed transport problem the asymptotic solution results in a diffusion approximation to the glbe

levy type diffusion on one dimensional directed cantor graphs levy type walks with correlated jumps induced by the topology of the medium are studied on a class of one dimensional deterministic graphs built from generalized cantor and smith volterra cantor sets the particle performs a standard random walk on the sets but is also allowed to move ballistically throughout the empty regions using scaling relations and the mapping onto the electric network problem we obtain the exact values of the scaling exponents for the asymptotic return probability the resistivity and the mean square displacement as a function of the topological parameters of the sets interestingly the system undergoes a transition from superdiffusive to diffusive behavior as a function of the filling of the fractal the deterministic topology also allows us to discuss the importance of the choice of the initial condition in particular we demonstrate that local and average measurements can display different asymptotic behavior the analytic results are compared to the numerical solution of the master equation of the process