α stable interference modeling and cauchy receiver for an ir uwb ad hoc network in this paper we investigate ad hoc networks based on impulse radio ultra wideband due to multiple access the interference distribution is not gaussian one important reason for errors is the presence of close interferers generating pulse collision however such events are rare and we propose an stable model compatible with this fact due to its heavy tailed distribution we derive the analytical expression of the two significant parameters they depend on the attenuation coefficient the users density the pulse collision probability and the pulse shape we finally propose receiver strategies cauchy receiver and p norm that outperforms the classical gaussian receiver

spectral correlations of fractional brownian motion fractional brownian motion fbm is a ubiquitous nonstationary model for many physical processes with power law time averaged spectra in this paper we exploit the nonstationarity to derive the full spectral correlation structure of fbm starting from the time varying correlation function we derive two different time frequency spectral correlation functions the ambiguity function and the kirkwood rihaczek spectrum and one dual frequency spectral correlation function the dual frequency spectral correlation has a surprisingly simple structure with spectral support on three discrete lines the theoretical predictions are verified by spectrum estimates of monte carlo simulations and of a time series of earthquakes with a magnitude of 7 and higher

a class of non gaussian processes for monte carlo simulation a new model is proposed to represent non gaussian stationary processes and develop monte carlo simulation algorithms for generating sample paths of non gaussian processes the model is based on a class of non gaussian processes the class of conditional gaussian processes two representations are considered for these processes the first representation is based on a randomized version of the classical spectral density function the second representation uses the output of a linear filter with random coefficients subjected to gaussian noise to define conditional gaussian processes the proposed model its representations and corresponding monte carlo simulation algorithms are illustrated by examples involving non gaussian random variables and processes it is shown that the proposed model can match any second moment properties but generally can only fit approximately a specified marginal distribution

performance analysis of ldpc coded diversity combining on rayleigh fading channels with impulsive noise spatial diversity is an effective method to mitigate the effects of fading and when used in conjunction with low density parity check ldpc codes it can achieve excellent error correcting performance noise added at each branch of the diversity combiner is generally assumed to be additive white gaussian noise but there are many applications where the received signal is impaired by noise with a non gaussian distribution in this paper we derive the exact bit error probability of different linear combining techniques on rayleigh fading channels with impulsive noise which is modeled using symmetric alpha stable distributions the relationship for the signal to noise ratios of these linear combiners is derived and then different non linear detectors are presented a detector based on the bi parameter cauchy gaussian mixture model is used and shows near optimal performance with a significant reduction in complexity when compared with the optimal detector furthermore the threshold signal to noise ratio of ldpc codes for different combining techniques on these channels is derived using density evolution and an estimation of the waterfall performance of ldpc codes is derived that reduces the gap between simulated and asymptotic performance

random walks at random times convergence to iterated levy motion fractional stable motions and other self similar processes for a random walk defined for a doubly infinite sequence of times we let the time parameter itself be an integer valued process and call the orginal process a random walk at random time we find the scaling limit which generalizes the so called iterated brownian motion khoshnevisan and lewis ann appl probab 9 1999 629 667 suggested the existence of a form of measure theoretic duality between iterated brownian motion and a brownian motion in random scenery we show that a random walk at random time can be considered a random walk in alternating scenery thus hinting at a mechanism behind this duality following cohen and samorodnitsky ann appl probab 16 2006 1432 1461 we also consider alternating random reward schema associated to random walks at random times whereas random reward schema scale to local time fractional stable motions we show that the alternating random reward schema scale to indicator fractional stable motions finally we show that one may recursively subordinate random time processes to get new local time and indicator fractional stable motions and new stable processes in random scenery or at random times when 2 the fractional stable motions given by the recursion are fractional brownian motions with dyadic h 0 1 also we see that un subordinating via a time change allows one to in some sense extract brownian motion from fractional brownian motions with h 1 2

particle tracking for fractional diffusion with two time scales abstract previous work y zhang m m meerschaert b baeumer particle tracking for time fractional diffusion phys rev e 78 2008 036705 showed how to solve time fractional diffusion equations by particle tracking this paper extends the method to the case where the order of the fractional time derivative is greater than one a subordination approach treats the fractional time derivative as a random time change of the corresponding cauchy problem with a first derivative in time one novel feature of the time fractional case of order greater than one is the appearance of clustering in the operational time subordinator which is non markovian solutions to the time fractional equation are probability densities of the underlying stochastic process the process models movement of individual particles the evolution of an individual particle in both space and time is captured in a pair of stochastic differential equations or langevin equations monte carlo simulation yields particle location and the ensemble density approximates the solution to the variable coefficient time fractional diffusion equation in one or several spatial dimensions the particle tracking code is validated against inverse transform solutions in the simplest cases further applications solve model equations for fracture flow and upscaling flow in complex heterogeneous porous media these variable coefficient time fractional partial differential equations in several dimensions are not amenable to solution by any alternative method so that the grid free particle tracking approach presented here is uniquely appropriate

heavy tailed log hydraulic conductivity distributions imply heavy tailed log velocity distributions 1 equations of contaminant transport describing non gaussian dispersion of solute in heterogeneous porous media have been developed by several authors e g berkowitz and scher 1995 1998 benson 1998 2001 berkowitz et al 2000 and baeumer et al 2005 the equations assume that solute particle movement in time and or space can be described by heavy tailed probability distributions e g levy stable distributions this assumption requires a mechanism for solute particles to move through heterogeneous porous media in a manner that can be approximated with heavy tailed probability distributions this work investigates whether a heavy tailed levy stable distribution of increments in log hydraulic conductivity results in a heavy tailed distribution of increments in log velocity a heavy tailed distribution of increments in log velocity would be a mechanism for solute particle movement in time or space that can be described by a heavy tailed probability distribution a computer algorithm was written to generate realistic hydraulic conductivity fields the algorithm modifies the spectral synthesis method for generating fractional brownian motion i e saupe 1988 and brewer and wheatcraft 1994 and produces self similar hydraulic conductivity fields with a heavy tailed levy stable distribution of increments in log hydraulic conductivity a monte carlo approach was adopted in which 100 hydraulic conductivity fields were generated for each of the following values of 0 8 1 1 1 4 and 1 7 the usgs finite difference groundwater code modflow was used to calculate the velocity fields and the increments in longitudinal log hydraulic conductivity and log velocity from the fields were analyzed to determine whether they were consistent with a heavy tailed levy stable probability density function pdf after determining that the distributions of increments in log hydraulic conductivity and log velocity were consistent with a levy stable pdf the tail parameters describing each pair of hydraulic conductivity and velocity fields were estimated with the nolan 1997 maximum likelihood estimator the relationship between the two tail parameters was then determined there are three conclusions of this research first mandelbrot and pp plots confirm that levy stable distributions of increments in log hydraulic conductivity give rise to levy stable distributions of increments in log velocity second the tail parameter for the increments in log velocity is typically smaller indicating a heavier probability tail than the corresponding tail parameter for the increments in log hydraulic conductivity third there is a positive statistical relation between the two tail parameters

simple random walk on long range percolation clusters ii scaling limits supported in part at the technion by a landau fellowship supported in part by an alfred sloan fellowship in mathematics

time changed ornstein uhlenbeck process the ornstein uhlenbeck process is one of the most popular systems used for financial data description however this process has also been examined in the context of many other phenomena in this paper we consider the so called time changed ornstein uhlenbeck process in which time is replaced by an inverse subordinator of general infinite divisible distribution time changed processes nowadays play an important role in various fields of mathematical physics chemistry and biology as well as in finance in this paper we examine the main characteristics of the time changed ornstein uhlenbeck process such as the covariance function moreover we also prove the formula for a generalized fractional fokker planck equation that describes the one dimensional probability density function of the analyzed system for three cases of subordinators we show the special forms of obtained general formulas furthermore we mention how to simulate the trajectory of the ornstein uhlenbeck process delayed by a general inverse subordinator

characteristic kernels and infinitely divisible distributions we connect shift invariant characteristic kernels to infinitely divisible distributions on rd characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures the contribution of this paper is twofold first we show using the levy khintchine formula that any shift invariant kernel given by a bounded continuous and symmetric probability density function pdf of an infinitely divisible distribution on rd is characteristic we mention some closure properties of such characteristic kernels under addition pointwise product and convolution second in developing various kernel mean algorithms it is fundamental to compute the following values i kernel mean values mp x x and ii kernel mean rkhs inner products mp mq h for probability measures p q if p q and kernel k are gaussians then the computation of i and ii results in gaussian pdfs that are tractable we generalize this gaussian combination to more general cases in the class of infinitely divisible distributions we then introduce a conjugate kernel and a convolution trick so that the above i and ii have the same pdf form expecting tractable computation at least in some cases as specific instances we explore stable distributions and a rich class of generalized hyperbolic distributions where the laplace cauchy and student s t distributions are included

harmonic oscillator under levy noise unexpected properties in the phase space a harmonic oscillator under the influence of noise is a basic model of various physical phenomena under gaussian white noise the position and velocity of the oscillator are independent random variables which are distributed according to the bivariate gaussian distribution with elliptic level lines the distribution of phase is homogeneous none of these properties hold in the general levy case thus the level lines of the joint probability density are not elliptic the coordinate and the velocity of the oscillator are strongly dependent and this dependence is quantified by introducing the corresponding parameter width deficit the distribution of the phase is inhomogeneous and highly nontrivial

a subspace based direction finding algorithm using fractional lower order statistics we propose several classes of fractional lower order moment flom based matrices that can be used with music to estimate the doas of independent circular signals embedded in additive s spl alpha s symmetric spl alpha stable noise e g sea clutter we run simulations with different choices of the flom parameter p for our flom based matrices and conclude that when the noise is s spl alpha s with unknown spl alpha spl ne 2 flom multiple signal classification music with p close to unity yields good performance the performance of flom music and robust covariation based roc music are similar three scenarios that contain circular signals phase modulation pm circularly symmetrical gaussian and quaternary phase shift keying qpsk and one scenario that contains noncircular signals binary phase shift keying bpsk all embedded in the same s spl alpha s noise are tested these simulation results reveal that the scenario containing bpsk signals leads to poor performance indicating that flom music is presently limited to circular signals

nonlinear system identification in impulsive environments nonlinear system identification has been studied under the assumption that the noise has finite second and higher order statistics in many practical applications impulsive measurement noise severely weakens the effectiveness of conventional methods in this paper spl alpha stable noise is used as a noise model in such case the minimum mean square error mmse criterion is no longer an appropriate metric for estimation error due to the lack of finite second order statistics of the noise therefore we adopt minimum dispersion criterion which in turn leads to the adaptive least mean pth power lmp algorithm it is shown that the lmp algorithm under the spl alpha stable noise model converges as long as the step size satisfies certain conditions the effect of p on the performance is also investigated compared with conventional methods the proposed method is more robust to impulsive noise and has better performance

ornstein uhlenbeck process with fluctuating damping abstract this paper studies langevin equation with random damping due to multiplicative noise and its solution two types of multiplicative noise namely the dichotomous noise and fractional gaussian noise are considered their solutions are obtained explicitly with the expressions of the mean and covariance determined explicitly properties of the mean and covariance of the ornstein uhlenbeck process with random damping in particular the asymptotic behavior are studied the effect of the multiplicative noise on the stability property of the resulting processes is investigated

density parameter estimation of skewed spl alpha stable distributions over the last few years there has been a great interest in spl alpha stable distributions for modeling impulsive data as a critical step in modeling with spl alpha stable distributions the problem of estimating the parameters of stable distributions have been addressed by several works in the literature however many of these works consider only the special case of symmetric stable random variables this is an important restriction however since most real life signals are skewed the existing techniques on estimating skewed distribution parameters are either computationally too expensive require lookup tables or have poor convergence properties we introduce three novel classes of estimators for the parameters of general stable distributions which are generalizations of the methods previously suggested for parameter estimation of symmetric stable distributions these estimators exploit expressions we develop for fractional lower order negative order and logarithmic moments and tail statistics we also introduce simple transformations that allow one to use existing symmetric stable parameter estimation techniques techniques suggested in this paper provide the only closed form solutions we are aware of for parameters that may be efficiently computed simulation results show that at least one of our new estimators has better performance than the existing techniques over most of the parameter space furthermore our techniques require substantially less computation

the gradually truncated levy flight for systems with power law distributions abstract power law distributions have been observed in various economical and physical systems levy flights have infinite variance which discourage a physical approach we introduce a class of stochastic processes the gradually truncated levy flight in which large steps of a levy flight are gradually eliminated it has finite variance and the system can be analyzed in a closed form we applied the present method to explain the distribution of a particular economical index the present method can be applied to describe time series in a variety of fields i e turbulent flow anomalous diffusion polymers etc

partly conditional estimation of the effect of a time dependent factor in the presence of dependent censoring we propose semiparametric methods for estimating the effect of a time dependent covariate on treatment free survival the data structure of interest consists of a longitudinal sequence of measurements and a potentially censored survival time the factor of interest is time dependent treatment free survival is of interest and is dependently censored by the receipt of treatment patients may be removed from consideration for treatment temporarily or permanently the proposed methods combine landmark analysis and partly conditional hazard regression a set of calendar time cross sections is specified and survival time from cross section date is modeled through weighted cox regression the assumed model for death is marginal in the sense that time varying covariates are taken as fixed at each landmark with the mortality hazard function implicitly averaging across future covariate trajectories dependent censoring is overcome by a variant of inverse probability of censoring weighting ipcw the proposed estimators are shown to be consistent and asymptotically normal with consistent covariance estimators provided simulation studies reveal that the proposed estimation procedures are appropriate for practical use we apply the proposed methods to pre transplant mortality among end stage liver disease esld patients

multiscaling fractional advection dispersion equations and their solutions 1 the multiscaling fractional advection dispersion equation ade is a multidimensional model of solute transport that encompasses linear advection fickian dispersion and super fickian dispersion the super fickian term in these equations has a fractional derivative of matrix order that describes unique plume scaling rates in different directions the directions need not be orthogonal so the model can be applied to irregular noncontinuum fracture networks the statistical model underlying multiscaling fractional dispersion is a continuous time random walk ctrw in which particles have arbitrary jump length distributions and finite mean waiting time distributions the meaning of the parameters in a compound poisson process a subset of ctrws is used to develop a physical interpretation of the equation variables the green s function solutions are the densities of operator stable probability distributions the limit distributions of normalized sums of independent and identically distributed random vectors these densities can be skewed heavy tailed and scale nonlinearly resembling solute plumes in granular aquifers they can also have fingers in any direction resembling transport along discrete pathways such as fractures

wavelet based parameter estimation for polynomial contaminated fractionally differenced processes we consider the problem of estimating the parameters for a stochastic process using a time series containing a trend component trend i e large scale variations in the series that are best modeled outside of a stochastic framework is often confounded with low frequency stochastic fluctuations this problem is particularly evident in models such as fractionally differenced fd processes which exhibit slowly decaying autocorrelations and can be extended to encompass nonstationary processes with substantial low frequency components we use the discrete wavelet transform dwt to estimate parameters for stationary and nonstationary fd processes in a model of polynomial trend plus fd noise using daubechies wavelet filters allows for automatic elimination of polynomial trends due to embedded differencing operations parameter estimation is based on an approximate maximum likelihood approach made possible by the fact that the dwt decorrelates fd processes approximately we consider this decorrelation in detail examining the between and within scale wavelet correlations separately better between scale decorrelation can be achieved by increasing the length of the wavelet filter whereas the within scale correlations can be handled via explicit modeling by a low order autoregressive process we demonstrate our methodology by applying it to a popular climate dataset

on the value distribution of the epstein zeta function in the critical strip we study the value distribution of the epstein zeta function e n l s for 0 infinity we prove that the random variable v n 2c e n cn has a limit distribution which we give explicitly here v n is the volume of the n dimensional unit ball more generally for any fixed epsilon 0 we determine the limit distribution of the random function c bar right arrow v n 2c e n cn c epsilon 1 4 epsilon 1 2 epsilon after compensating for the pole at c 1 2 we even obtain a limit result on the whole interval 1 4 epsilon 1 2 and as a special case we deduce the following strengthening of a result by sarnak and strombergsson concerning the height function h n l of the flat torus r n l the random variable n h n l log 4 pi gamma 1 log n has a limit distribution as n infinity which we give explicitly finally we discuss a question posed by sarnak and strombergsson as to whether there exists a lattice l subset of r n for which e n l s has no zeros in 0 infinity

superdiffusive limits for deterministic fast slow dynamical systems we consider deterministic fast slow dynamical systems on mathbb r m times y of the form begin aligned left begin array ll x k 1 n x k n n 1 a big x k n big n 1 alpha b big x k n big v y k y k 1 f y k end array right end aligned where alpha in 1 2 under certain assumptions we prove convergence of the m dimensional process x n t x lfloor nt rfloor n to the solution of the stochastic differential equation begin aligned mathrm d x a x mathrm d t b x diamond mathrm d l alpha end aligned where l alpha is an alpha stable levy process and diamond indicates that the stochastic integral is in the marcus sense in addition we show that our assumptions are satisfied for intermittent maps f of pomeau manneville type

the method of simulated quantiles we introduce an inference method based on quantiles matching which is useful for situations where the density function does not have a closed form but it is simple to simulate and or moments do not exist functions of theoretical quantiles which depend on the parameters of the assumed probability law are matched with sample quantiles which depend on observations since the theoretical quantiles may not be available analytically the optimization is based on simulations we illustrate the method with the estimation of stable distributions a thorough monte carlo study and an illustration to 22 nancial indexes show

stability and lack of memory of the returns of the hang seng index abstract in this paper we show that the logarithmic returns of the hang seng index from january 2 1987 to november 14 2005 statistically resemble a sequence of independent identically distributed levy stable random variables this is in stark contrast to xiu and jin 2007 39 where long memory farima processes with gaussian noise were suggested as well fitted to the data the lack of memory is checked by using lo s modified r s statistic and a new method of estimation of the memory parameter d which applies the notion of empirical mean squared displacement in order to test stability of the data we employ several statistical tests based on the empirical distribution function finally we also show that the returns possess no conditional heteroscedasticity property thus excluding the arch garch family of processes as possible underlying models

noise induced escape in an excitable system we consider the stochastic dynamics of escape in an excitable system the fitzhugh nagumo fhn neuronal model for different classes of excitability we discuss first the threshold structure of the fhn model as an example of a system without a saddle state we then develop a nonlinear nonlocal stability approach based on the theory of large fluctuations including a finite noise correction to describe noise induced escape in the excitable regime we show that the threshold structure is revealed via patterns of most probable optimal fluctuational paths the approach allows us to estimate the escape rate and the exit location distribution we compare the responses of a monostable resonator and monostable integrator to stochastic input signals and to a mixture of periodic and stochastic stimuli unlike the commonly used local analysis of the stable state our nonlocal approach based on optimal paths yields results that are in good agreement with direct numerical simulations of the langevin equation

near derivations in lie algebras abstract let l be a lie algebra we call a linear map f l l a near derivation if there exists a linear map g l l such that ad x f g ad x is a derivation for every x l the paper is devoted to describing the structure of near derivations in certain lie algebras arising from associative ones

an rkhs formulation of the inverse regression dimension reduction problem suppose that y is a scalar and x is a second order stochastic process where y and x are conditionally independent given the random variables 1 p which belong to the closed span l2 of x this paper investigates a unified framework for the inverse regression dimension reduction problem it is found that the identification of l2 with the reproducing kernel hilbert space of x provides a platform for a seamless extension from the finite to infinite dimensional settings it also facilitates convenient computational algorithms that can be applied to a variety of models

on bayesian modeling of fat tails and skewness abstract we consider a bayesian analysis of linear regression models that can account for skewed error distributions with fat tails the latter two features are often observed characteristics of empirical datasets and we formally incorporate them in the inferential process a general procedure for introducing skewness into symmetric distributions is first proposed even though this allows for a great deal of flexibility in distributional shape tail behavior is not affected applying this skewness procedure to a student t distribution we generate a skewed student distribution which displays both flexible tails and possible skewness each entirely controlled by a separate scalar parameter the linear regression model with a skewed student error term is the main focus of the article we first characterize existence of the posterior distribution and its moments using standard improper priors and allowing for inference on skewness and tail parameters for posterior inference with this model we suggest

bayesian likelihood methods for estimating the end point of a distribution summary we consider maximum likelihood methods for estimating the end point of a distribution the likelihood function is modified by a prior distribution that is imposed on the location parameter the prior is explicit and meaningful and has a general form that adapts itself to different settings results on convergence rates and limiting distributions are given in particular it is shown that the limiting distribution is non normal in non regular cases parametric bootstrap techniques are suggested for quantifying the accuracy of the estimator we illustrate performance by applying the method to multiparameter weibull and gamma distributions

wavelet entropy and fractional brownian motion time series abstract we study the functional link between the hurst parameter and the normalized total wavelet entropy when analyzing fractional brownian motion fbm time series these series are synthetically generated both quantifiers are mainly used to identify fractional brownian motion processes l zunino d g perez m garavaglia o a rosso characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform fractals 12 2 2004 223 233 the aim of this work is to understand the differences in the information obtained from them if any

memoryless discrete time signal detection in long range dependent noise the problem of designing optimum memoryless detectors for known signals in long range dependent lrd noise is considered in particular under the performance criterion of asymptotic relative efficiency are optimum memoryless detection in lrd noise is investigated by exploiting the hermite expansions the detectors considered have the form of a nonlinearity followed by an accumulator and threshold comparator it is shown that when the noise is lrd and gaussian all nonlinearities with hermite rank one are asymptotically equivalent in terms of efficiency and are most powerful moreover the optimum nonlinearities with hermite rank greater than one are given by the corresponding hermite polynomials the case of non gaussian lrd noise that can be derived by nonlinear transformation of gaussian noise is also considered in this case the globally optimum nonlinearity is very difficult to obtain in general instead we proposed a suboptimal nonlinearity which is given by a linear combination of the corresponding locally optimum detector and the inverse of the transform function generating the noise simulations show that the proposed detector outperforms the locally optimal detector for lrd non gaussian noise

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

tempered stable levy motion driven by stable subordinator a b s t r a c t in this article we propose a new model for financial data description combining two independent mechanisms namely the tempered stable process and inverse stable subordinator we obtain a new model which captures not only the tempered stable character of the underlying data but also such a property as periods in which the values of an asset stay on the same level moreover we classify our system to the family of subdiffusive processes and investigate its tail behavior we describe in detail testing and estimation procedures for the proposed model in the last step we calibrate our model to the real data

erratum to a stochastic controller for a scalar linear system with additive cauchy noise automatica 50 1 2014 114 127 abstract a stochastic control scheme is developed for scalar discrete time and linear dynamic systems driven by cauchy distributed process and measurement noises when addressing the optimal control problem for such systems the standard quadratic cost criteria cannot be used in this study we introduce a new objective function that is functionally similar to the cauchy probability density function the performance index defined as the expectation of this objective function with respect to the cauchy densities exists the dynamic programming solution to the fixed and finite horizon optimal control problem that uses this performance index appears to be intractable therefore a moving horizon optimal model predictive control problem is implemented for which the conditional expected value of the objective function and its gradients can be computed in closed form and without assumptions such as certainty equivalence numerical results are shown for this m step model predictive optimal controller and compared to a similar linear exponential gaussian model predictive controller an essential difference between the cauchy and gaussian controllers when applied to a system with cauchy noises is that while the gaussian controller is linear and reacts strongly to all noise pulses the cauchy controller can differentiate between measurement and process noise pulses by ignoring the former while responding to the latter this property of the cauchy controller occurs when an impulsive measurement noise is more likely than an impulsive process noise the cauchy and gaussian controllers react similarly when applied to a system with gaussian noises demonstrating the robustness of the proposed control scheme

zero order statistics a mathematical framework for the processing and characterization of very impulsive signals impulsive or heavy tailed processes with infinite variance appear naturally in a variety of practical problems that include wireless communications teletraffic hydrology geology and economics most signal processing and statistical methods available in the literature have been designed under the assumption that the processes possess finite variance and they usually break down in the presence of infinite variance although methods based on fractional lower order statistics flos have proven successful in dealing with infinite variance processes they fail in general when the noise distribution has very heavy algebraic tails in this paper we introduce a new approach to statistical moment characterization which is well defined over all processes with algebraic or lighter tails unlike flos these zero order statistics zos as we will call them provide a common ground for the analysis of basically any distribution of practical use known today three new parameters namely the geometric power the zero order location and the zero order dispersion constitute the foundation of zos they play roles similar to those played by the power the expected value and the standard deviation in the theory of second order processes we analyze the properties of the new parameters and derive a zos framework for location estimation that gives rise to a novel mode type estimator with important optimality properties under very impulsive noise several simulations are presented to illustrate the potential of zos methods

the stochastic geometry analyses of cellular networks with alpha stable self similarity to understand the spatial deployment of base stations bss is the first step to analyze the performance of cellular networks and further design efficient networking protocols poisson point process ppp which has been widely adopted to characterize the deployment of bss and established the reputation to give tractable results in the stochastic geometry analyses usually assumes a static bs deployment density in homogeneous ppp hppp models or delicately designed location dependent density functions in in homogeneous ppp models however the simultaneous existence of attractiveness and repulsiveness among bss practically deployed in a large scale area defies such an assumption and the alpha stable distribution one kind of heavy tailed distributions has recently demonstrated superior accuracy to statistically model the varying bs density in different areas in this paper we start with these new findings and investigate the intrinsic feature i e the spatial self similarity embedded in the bss afterwards we refer to a generalized ppp setup with alpha stable distributed density and theoretically derive the related coverage probability in particular we give an upper bound of the derived coverage probability for high signal to interference plus noise ratio thresholds and show the monotonically decreasing property of this bound with respect to the variance of bs density besides we prove that our model could reduce to the single tier hppp for some special cases and demonstrate the superior accuracy of the alpha stable model to approach the real environment

on stochastic differential equations driven by a cauchy process and other stable levy motions we consider the class of one dimensional stochastic differential equations dx t b x t dz t t 0 where b is a borel measurable real function and z is a strictly stable levy process 0 2 weak solutions are investigated improving previous results of the author in various ways in particular for the equation driven by a strictly 1 stable levy process a sufficient existence condition is proven also we extend the weak existence and uniqueness exact criteria due to engelbert and schmidt for the brownian case i e a 2 to the class of equations with a such that 1 2 the results employ some representation properties with respect to strictly stable levy processes

a multivariate distribution for sub images a new method for obtaining multivariate distributions for sub images of natural images is described the information in each sub image is summarized by a measurement vector in a measurement space the dimension of the measurement space is reduced by applying a random projection to the truncated output of the discrete cosine transforms of the sub images the measurement space is then reparametrized such that a gaussian distribution is a good model for the measurement vectors in the reparametrized space an ornstein uhlenbeck process associated with the gaussian distribution is used to model the differences between measurement vectors obtained from matching sub images the probability of a false alarm and the probability of accepting a correct match are calculated the accuracy of the resulting statistical model for matching sub images is tested using images from the middlebury stereo database with promising results in particular if the probability of accepting a correct match is relatively large then there is good agreement between the calculated and the experimental probabilities of obtaining a unique match that is also a correct match

stable random variables convolution and reliability in this work first we obtain a representation of the convolution between two asymmetric stable random variables of different parameters in terms of the fox h function in the symmetric case we obtain in terms of the meijer g function a computable expression of the convolution for rational stability indices physically this density function can be interpreted as the probabilistic generalization of the voigt profile function second we obtain the reliability p x y for symmetric stable random variables of different stability indices scale and shift parameters graphical illustrations and error analysis are given for the convolution and the reliability is tested by monte carlo simulation

nonparametric reconstruction of a multifractal function from noisy data we estimate a real valued function f of d variables subject to additive gaussian perturbation at noise level varepsilon 0 under l loss for 1 the main novelty is that f can have an extremely varying local smoothness exhibiting a so called multifractal behaviour the results of jaffard on the frisch parisi conjecture suggest to link the singularity spectrum of f to besov properties of the signal that can be handled by wavelet thresholding for denoising purposes we prove that the optimal minimax rate of estimation of multifractal functions with singularity spectrum d h has explicit representation varepsilon 2v d bullet pi with v d bullet pi min h frac h left d d h right pi 2h d the minimum is taken over a specific domain and the rate is corrected by logarithmic factors in some cases in particular the usual rate varepsilon 2s 2s d is retrieved for monofractal functions with spectrum reduced to a single value s irrespectively of more interestingly the sparse case of estimation over single besov balls has a new interpretation in terms of multifractal analysis

on the scaling of the distribution of daily price fluctuations in the mexican financial market index abstract in this paper a statistical analysis of log return fluctuations of the ipc the mexican stock market index is presented a sample of daily data covering the period from 04 09 2000 04 09 2010 was analyzed and fitted to different distributions tests of the goodness of fit were performed in order to quantitatively asses the quality of the estimation special attention was paid to the impact of the size of the sample on the estimated decay of the distributions tail in this study a forceful rejection of normality was obtained on the other hand the null hypothesis that the log fluctuations are fitted to a stable levy distribution cannot be rejected at the 5 significance level

data driven rank tests for independence we introduce new rank tests for testing independence the new testing procedures are sensitive not only for grade linear correlation but also for grade correlations of higher order polynomials the number of polynomials involved is determined by the data model selection is combined with application of the score test in the selected model whereas well known tests as spearman s test or hoeffding s test may completely break down for alternatives that are dependent but have low grade linear correlation the new tests have greater power stability monte carlo results clearly show this behavior theoretical support is obtained by proving consistency of the new tests

gini estimation under infinite variance we study the problems related to the estimation of the gini index in presence of a fat tailed data generating process i e one in the stable distribution class with finite mean but infinite variance i e with tail index 1 2 we show that in such a case the gini coefficient cannot be reliably estimated using conventional nonparametric methods because of a downward bias that emerges under fat tails this has important implications for the ongoing discussion about economic inequality we start by discussing how the nonparametric estimator of the gini index undergoes a phase transition in the symmetry structure of its asymptotic distribution as the data distribution shifts from the domain of attraction of a light tailed distribution to that of a fat tailed one especially in the case of infinite variance we also show how the nonparametric gini bias increases with lower values of we then prove that maximum likelihood estimation outperforms nonparametric methods requiring a much smaller sample size to reach efficiency finally for fat tailed data we provide a simple correction mechanism to the small sample bias of the nonparametric estimator based on the distance between the mode and the mean of its asymptotic distribution

space fractional advection dispersion equations with variable parameters diverse formulas numerical solutions and application to the macrodispersion experiment site data 1 to model the observed local variation of transport speed an extension of the homogeneous space fractional advection dispersion equation fade to more general cases with space dependent coefficients drift velocity v and dispersion coefficient d has been suggested to provide a rigorous evaluation of this extension we explore the underlying physical meanings of two proposed and one other possible form of the fade by using the generalized mass balance law proposed by meerschaert et al 2006 when the classical fick s law is replaced by its generalized form in the first order mass conversation law the original fade with constant parameters extends to the advection dispersion equation ade with fractional flux denoted as ff ade when the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence we get the ade with fractional divergence fd ade when the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form the fade contains fully fractional divergence ffd ade these three fades with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by cushman et al 1994 while the space variable fades correspond to the conditional nonlocal theory proposed by neuman 1993 after specifying the general levy type functional form of the random distribution the corresponding langevin markov models can be found in many cases where the lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical measurable level and resolution the resulting lagrangian random walk particle tracking methods along with previous numerical solutions using implicit euler finite differences distinguish and elucidate the plume behavior described by these fades the fractional models are applied to fit the tritium plumes measured at the macrodispersion experiment test site when the local parameters gleaned from the hydraulic conductivity k distribution are varied even slightly i e a two zone model allowing the use of observed hydraulic conditions at the site the model fits are well within the variability of the data the extended fades describe the fast and space dependent leading edges of measured plumes in the regional scale alluvial system which was underestimated and could not be fully captured by the original fade with constant parameters applications also favor the fd ade model because of the ease of implementation and consistency with previous analysis of the k statistics

soft decision metrics for coded orthogonal signaling in symmetric alpha stable noise this paper derives new soft decision metrics for coded orthogonal signaling in impulsive noise more specifically symmetric alpha stable noise for the case of a known channel amplitude and known noise dispersion exact metrics are derived both for cauchy and gaussian noise for the case that the channel amplitude or the dispersion is unknown approximate metrics are obtained in closed form based on a generalized likelihood ratio approach the performance of the new metrics is compared numerically for a turbo coded system and the sensitivity to side information of the optimum receiver for cauchy noise is considered the gain that can be achieved by using a properly chosen decoding metric is quantified and it is shown that this gain is significant the application of the results to frequency hopping ad hoc networks is also discussed

modeling of rainfall time series and extremes using bounded random cascades and levy stable distributions a new model for simulation of rainfall time series is proposed it is shown that both the intensity and duration of individual rainfall events can be best modeled by a fat tailed levy stable distribution the temporal downscaling of individual events is produced by a new type of a bounded random cascade model the proposed rainfall model is shown to successfully reproduce the statistical behavior of individual storms as well as and in particular the statistical behavior of annual maxima in contrast a model based on a gamma distribution for rainfall intensity substantially underestimates the absolute values of extreme events and does not correctly reproduce their scaling behavior similarly a model based on self similar random cascade as opposed to the bounded cascade substantially overestimates the extreme events

the spectrum of heavy tailed random matrices let x n be an n n random symmetric matrix with independent equidistributed entries if the law p of the entries has a finite second moment it was shown by wigner 14 that the empirical distribution of the eigenvalues of x n once renormalized by sqrt n converges almost surely and in expectation to the so called semicircular distribution as n goes to infinity in this paper we study the same question when p is in the domain of attraction of an stable law we prove that if we renormalize the eigenvalues by a constant a n of order n frac 1 alpha the corresponding spectral distribution converges in expectation towards a law mu alpha which only depends on we characterize mu alpha and study some of its properties it is a heavy tailed probability measure which is absolutely continuous with respect to lebesgue measure except possibly on a compact set of capacity zero

characteristic functions of random variables attracted to 1 stable laws the domain of attraction of a 1 stable law on r d is characterized by the expansions of the characteristic functions of its elements

moment analysis for spatiotemporal fractional dispersion 1 the evolution of the first five nonnegative integer order spatial moments corresponding to the mass mean variance skewness and kurtosis are investigated systematically for spatiotemporal nonlocal fractional dispersion three commonly used fractional order transport equations including the time fractional advection dispersion equation time fade the fractal mobile immobile mim equation and the fully fractional advection dispersion equation ffade are considered analytical solutions verify our numerical results and reveal the anomalous evolution of the moments following adams and gelhar s 1992 work on the classical ade we find that a simultaneous analysis of all moments is critical in discriminating between different nonlocal models the evolution of dispersion among the subdiffusive to superdiffusive rates is then further explored numerically by a non markovian random walk particle tracking method that can be used for any heterogeneous boundary or initial value problem in three dimensions both the analytical and the numerical results also show the similarity at the early time and the difference at the late time of moment growth for solutes in different phases mobile versus total described by the mim models further simulations of the 1 d bromide snapshots measured at the made experiments using all three models with parameters fitted by the observed zeroth to fourth moments indicate that 1 both the time and space nonlocality strongly affect the solute transport at the made site 2 all five spatial moments should be considered in transport model selection and calibration because those up to the variance cannot effectively discriminate between nonlocal models and 3 the log concentration should be used when evaluating the plume leading edge and the effects of space nonlocality

the jacobi and gauss seidel type iteration methods for the matrix equation a x b c in this paper the jacobi and gauss seidel type iteration methods are proposed for solving the matrix equation a x b c which are based on the splitting schemes of the matrices a and b the convergence and computational cost of these iteration methods are discussed furthermore we give the preconditioned jacobi and gauss seidel type iteration methods numerical examples are given to demonstrate the efficiency of these methods proposed in this paper

option pricing beyond black scholes based on double fractional diffusion abstract we show how the prices of options can be determined with the help of double fractional differential equation in such a way that their inclusion in a portfolio of stocks provides a more reliable hedge against dramatic price drops than the use of options whose prices were fixed by the black scholes formula

darling erdős theorems for normalized sums of i i d variables close to a stable law let 1 be i i d real valued random variables and s n 1 n in the case when the distribution of is close to a stable law for some 0 1 1 2 we investigate the asymptotic behavior in distribution of the maximum of normalized sums max k 1 k 1 s k this completes the darling erdos limit theorem for the case 2