stochastic flow cascades we introduce and explore a stochastic flow cascade sfc model a general statistical model for the unidirectional flow through a tandem array of heterogeneous filters examples include the flow of i liquid through heterogeneous porous layers ii shocks through tandem shot noise systems iii signals through tandem communication filters the sfc model combines together the langevin equation convolution filters and moving averages and poissonian randomizations a comprehensive analysis of the sfc model is carried out yielding closed form results levy laws are shown to universally emerge from the sfc model and characterize both heavy tailed retention times noah effect and long ranged correlations joseph effect

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

power law shot noise the behavior of power law shot noise for which the associated impulse response functions assume a decaying power law form is explored expressions are obtained for the moments moment generating functions amplitude probability density functions autocorrelation functions and power spectral densities for a variety of parameters of the process for certain parameters the power spectral density exhibits 1 f type behavior over a substantial range of frequencies so that the process serves as a source of 1 f sup alpha shot noise for alpha in the range 0

shot noise systems with random relaxations we explore a model of a random relaxation shot noise system in which i the shot inflow is a general poisson point process with possibly infinite poissonian rates and ii the exponential relaxations governing the shot decay are randomized this system model is applicable to physical environments polluted by radioactive contamination of heterogeneous types the statistics of random relaxation shot noise systems are analyzed quantitatively and comprehensively stationary structure correlation structure process distribution fractality and asymptotic fractality and the display both separately and simultaneously of the noah and joseph effects results are obtained explicitly and in closed form and facilitate the design of tractable shot noise systems with unique features

on the stability of static poisson networks under random access we investigate the stable packet arrival rate region of a discrete time slotted random access network where the sources are distributed as a poisson point process each of the sources in the network has a destination at a given distance and a buffer of infinite capacity the network is assumed to be random but static i e the sources and the destinations are placed randomly and remain static during all the time slots we employ tools from queueing theory as well as point process theory to study the stability of this system using the concept of dominance the problem is an instance of the interacting queues problem further complicated by the poisson spatial distribution we obtain sufficient conditions and necessary conditions for stability numerical results show that the gap between the sufficient conditions and the necessary conditions is small when the access probability the density of transmitters or the sinr threshold is small the results also reveal that a slight change of the arrival rate may greatly affect the fraction of unstable queues in the network

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

quartic splines solutions for a system of second order boundary value problems we use uniform quartic polynomial splines to develop a numerical method for computing smooth approximations to the solution and its derivatives for a system of second order boundary value problems associated with obstacle unilateral and contact problems we show that the present method gives approximations which are better than those produced by other collocation finite difference and spline methods

the use of cubic splines in the numerical solution of a system of second order boundary value problems abstract we use uniform cubic polynomial splines to develop some consistency relations which are then used to develop a numerical method for computing smooth approximations to the solution and its derivatives for a system of second order boundary value problems associated with obstacle unilateral and contact problems we show that the present method gives approximations which are better than those produced by other collocation finite difference and spline methods

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

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

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

gaussian approximation for the wireless multi access interference distribution the problem of gaussian approximation for the wireless multi access interference distribution in large spatial wireless networks is addressed first a principled methodology is presented to establish rates of convergence of the multi access interference distribution to a gaussian distribution for general bounded and power law decaying path loss functions the model is general enough to also include various random wireless channel dynamics such as fading and shadowing arising from multi path propagation and obstacles existing in the communication environment it is shown that the wireless multi access interference distribution converges to the gaussian distribution with the same mean and variance at a rate 1 where 0 is a parameter controlling the intensity of the planar possibly non stationary poisson point process generating node locations an explicit expression for the scaling coefficient is obtained as a function of fading statistics and the path loss function second an extensive numerical and simulation study is performed to illustrate the accuracy of the derived gaussian approximation bounds a good statistical fit between the interference distribution and its gaussian approximation is observed for moderate to high values of

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

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

fractal renewal processes two relatively simple renewal processes whose power spectral densities vary as 1 f sup d are constructed 1 a standard renewal point process with 0

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

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

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

spectral representation of fractional brownian motion in n dimensions and its properties fractional brownian motion fbm provides a useful model for processes with strong long term dependence such as 1 f sup spl beta spectral behavior however fbm s are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation to facilitate the study of this problem another model is provided for the construction of fbm from a white noise like process by means of a stochastic or ito integral in frequency of a stationary uncorrelated random process also a generalized power spectrum of the nonstationary fbm process is defined this new approach to fbm can be used to compute all of the correlations power spectra and other properties of fbm in this paper a number of these fbm properties are developed from this model such as the t sup h law of scaling the power law of fractional order the correlation of two arbitrary fbm s and the evaluation of the fractal dimension under various transformations this new treatment of fbm using a spectral representation is extended also for the first time to two or more topological dimensions in order to analyze the features of isotropic n dimensional fbm

resurgence flows in porous media porous media with resurgences can be described by a double structure namely a continuous porous medium and capillaries with impermeable walls which relate distant points of the continuous medium the resurgences can be either punctual or extended the equations for flow in such media are derived some general properties of the resulting system which involves nonlocal aspects are deduced a dilute approximation is detailed for punctual resurgences in two dimensional media and is illustrated by a few examples

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

shot noise displaying simultaneously the noah and joseph effects shot noise processes displaying the noah effect are of infinite variance hence the measurement of the correlation structure of such processes indicating whether the joseph effect is displayed as well cannot be conducted via their autocovariance functions which are undefined to circumvent this problem a poisson based analysis applicable to shot noise processes with infinite variance as well as to shot noise processes with divergent noise levels is developed a poissonian autocovariance function which characterizes the process distribution of general shot noise processes is introduced in particular this function governs and quantifies the processes stationary structure amplitudal behavior and correlation structure temporal behavior the poissonian methodology developed enables a precise quantitative analysis of shot noise processes displaying simultaneously the noah and joseph effects

the stability test for symmetric alpha stable distributions symmetric alpha stable distributions are a popular statistical model for heavy tailed phenomena encountered in communications radar biomedicine and econometrics the use of the symmetric alpha stable model is often supported by empirical evidence where qualitative criteria are used to judge the fit leading to subjective decisions objective decisions can only be made through quantitative statistical tests here a goodness of fit hypothesis test for symmetric alpha stable distributions is developed based on their unique stability property critical values for the test are found using both asymptotic theory and from bootstrap estimates experiments show that the stability test using bootstrap estimates of the critical values is better able to discriminate between symmetric alpha stable distributions and other heavy tailed distributions than classical tests such as the kolmogorov smirnov test

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

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

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

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

joint temporal statistics of interference in decentralized wireless networks characterizing interference statistics is central to the design and analysis of both physical layer and medium access control layer techniques to mitigate interference in a wireless network the applicability of interference statistics however is limited by the assumptions adopted to derive the statistics in closed form common assumptions for a decentralized wireless network include temporally independent user locations and an unbounded pathloss function in this correspondence we derive the joint temporal statistics of interference that capture the temporal correlation in the network along with the realistic assumption of a bounded pathloss function the closed form statistics are asymptotically exact for low tail probabilities and match closely in simulations even when the tail probability is fairly high the primary contributions are to i show that joint interference statistics follow a multivariate gaussian mixture distribution under the assumption of a bounded pathloss function and ii characterize the joint tail probability decay behavior for both bounded and unbounded pathloss functions

modeling heterogeneous network interference using poisson point processes cellular systems are becoming more heterogeneous with the introduction of low power nodes including femtocells relays and distributed antennas unfortunately the resulting interference environment is also becoming more complicated making evaluation of different communication strategies challenging in both analysis and simulation leveraging recent applications of stochastic geometry to analyze cellular systems this paper proposes to analyze downlink performance in a fixed size cell which is inscribed within a weighted voronoi cell in a poisson field of interferers a nearest out of cell interferer out of cell interferers outside a guard region and cross tier interferers are included in the interference calculations bounding the interference power as a function of distance from the cell center the total interference is characterized through its laplace transform an equivalent marked process is proposed for the out of cell interference under additional assumptions to facilitate simplified calculations the interference distribution is approximated using the gamma distribution with second order moment matching the gamma approximation simplifies calculation of the success probability and average rate incorporates small scale and large scale fading and works with co tier and cross tier interference simulations show that the proposed model provides a flexible way to characterize outage probability and rate as a function of the distance to the cell edge

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

fractal renewal processes generate 1 f noise 1 mathit f mathit d noise occurs in an impressive variety of physical systems and numerous complex theories have been proposed to explain it we construct two relatively simple renewal processes whose power spectral densities vary as 1 mathit f mathit d i a standard renewal point process with 0d1 and ii a finite valued alternating renewal process with 0d2 the resulting event number statistics coincidence rates minimal coverings and autocorrelation functions are shown also to follow power law forms these fractal characteristics derive from interevent time probability density functions which themselves decay in a power law fashion a number of applications are considered trapping in amorphous semiconductors electronic burst noise movement in systems with fractal boundaries the digital generation of 1 mathit f mathit d noise and ionic currents in cell membranes

multiple time scale analysis of macrotransport processes abstract convective and diffusive transport of a brownian tracer corpuscle is analyzed within a multidimensional space decomposed into local internal and global external subspaces multiple time scale methods are employed to successively eliminate from the kinetic equation governing the multidimensional microtransport process its dependence upon each of the internal coordinates the resulting long time equation describing the residual global space macrotransport process derived by this systematic perturbation procedure is shown to accord with the well established results of generalized taylor dispersion theory heretofore obtained by ad hoc arguments by way of example this macrotransport description is used to analyze the taylor dispersion of a solute in a poiseuille type solvent flow occuring within a rectangular duct of small but non zero aspect ratio elementary dispersion results obtained by ignoring the side walls apply only for relatively short times for longer times the perturbing presence of the side walls acts to substantially increase the axial dispersivity even in the limit of zero aspect ratio where intuition would strongly suggest otherwise

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

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

phase modulated resonances modeled as self similar processes with application to turbulent sounds in this paper we propose a nonlinear stochastic model for time varying resonances where the instantaneous phase and frequency of a sinusoidal oscillation is allowed to vary proportionally to an spl alpha stable self similar stochastic processes the main motivation of our work stems from previous experimental and theoretical evidence that speech resonances in fricative sounds can be modeled phenomenologically as am fm signals with randomly varying instantaneous frequencies and that several signal classes related to turbulent phenomena are self similar 1 f processes our general approach is to model the instantaneous phase of an am fm resonance as a self similar spl alpha stable process as a special case this random phase model includes the class of random fractal signals known as fractional brownian motion we theoretically explore this random modulation model and analytically derive its autocorrelation and power spectrum we also propose an algorithm to fit this model to arbitrary resonances with random phase modulation further we apply the above ideas to real speech data and demonstrate that this model is suitable for resonances of fricative sounds

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

power law shot noise and its relationship to long memory spl alpha stable processes we consider the shot noise process whose associated impulse response is a decaying power law kernel of the form t sup spl beta 2 1 we show that this power law poisson model gives rise to a process that at each time instant is an spl alpha stable random variable if spl beta 1 the power law poisson process has a power law spectrum we show that although in the case spl beta 1 the power spectrum does not exist the process still exhibits long memory in a generalized sense the power law shot noise process appears in many applications in engineering and physics the proposed results can be used to study such processes as well as to synthesize a random process with long range dependence

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

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

generalized taylor aris dispersion in discrete spatially periodic networks microfluidic applications a theory is presented for the lumped parameter convective diffusive transport of individual noninteracting brownian solute particles macromolecules moving within spatially periodic solvent filled networks the latter representing models of chip based microfluidic chromatographic separation devices as well as porous media using graph theoretical techniques the composite medium is conceptually decomposed into a network of channels the edges through which the solute is transported by a combination of molecular diffusion and either piggyback entrainment within a flowing solvent or an externally applied force field acting upon the solute molecules a probabilistic choice of egress channel for a solute particle exiting the intersection vertex of the channels is furnished by an imperfect mixing model a spatially periodic taylor aris like method of moments scheme is applied to this transport model leading to discrete matrix equations for computing the network scale particle velocity vector u and dispersivity dyadic d in terms of the prescribed microscale transport parameters and network geometry characterizing the basic unit cell of which the spatially periodic device is comprised the ensuing algebraic equations governing the vertex based discrete unit cell fields p 0 infinity i and b i i 1 2 n whose paradigmatic summations yield u and d constitute discrete analogs of classical continuous macrotransport phenomenological parameters p 0 infinity r and b r with r a continuous position vector defined within the unit cell the ease with which these discrete calculations can be performed for complex networks renders feasible parametric studies of potential microfluidic chip designs particularly those pertinent to biomolecular separation schemes application of this discrete theory to the dispersion analysis of pressure driven flow in spatially periodic serpentine microchannels is shown to accord with existing results previously derived using classical continuous macrotransport theory

fractal levy correlation cascades the correlation structure of a wide class of random processes driven by stable non gaussian levy noise sources is explored since these noises are of infinite variance correlations cannot be measured via auto covariance functions exploiting the underlying poissonian structure of levy noises we present a cascade of poissonian correlation functions which characterize the correlation structure and the process distribution of the processes under consideration the theory developed is applied to various examples including motions ornstein uhlenbeck and moving average processes and fractional motions and noises all driven by stable non gaussian levy noises

intrinsic fractality of classic shot noise we revisit the classic shot noise of campbell and schottky char22 a stochastic process governed by ornstein uhlenbeck dynamics driven by a poissonian noise exploring the order statistics of the shot magnitudes composing its stationary noise level we show that classic shot noise is intrinsically fractal this fractality is manifested by mathbf i intrinsic paretian and scale invariant structures mathbf ii an intrinsic power law scaling mathbf iii an intrinsic statistical resilience to random power law perturbations

some aspects of variational inequalities abstract in this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence uniqueness computational properties various generalizations sensitivity analysis and their applications we also propose some open problems with sufficient information and references so that someone may attempt solution s in his her area of special interest we also include some new results which we have recently obtained

brief box jenkins continuous time modeling this paper treats the identification of continuous time models using arbitrary band limited excitation signals a modeling approach is presented that has the following two advantages 1 asymptotically the amount of data tends to infinity there is no approximation error over the complete frequency band from dc to nyquist 2 it allows to identify general parametric noise models the key idea is to combine a continuous time plant model with a discrete time noise model hybrid box jenkins model structure

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

non linear shot noise levy noah joseph abstract we introduce and study a generic non linear shot noise system model shots of random magnitudes arrive to the system stochastically following an arbitrary time homogeneous poisson point process after hitting the system the magnitude of an arriving shot decays to zero the decay is governed by an arbitrary differential equation dynamics shots are independent and their overall effect on the system is additive the system s noise level at time t equals the sum of the magnitudes at time t of all the shots arriving to the system prior to time t the resulting shot noise is i a levy process when the decay dynamics are degenerate ii a levy driven ornstein uhlenbeck process when the decay dynamics are linear and iii a stationary non markov process when the decay dynamics are non linear the resulting shot noise admits an underlying levy structure which we explicitly compute and can yield both the noah effect and the joseph effect closed form analytic formulae for various statistics are derived including the log laplace transform and cumulants of the stationary noise level the process auto covariance function and the process range of dependence

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

signal modeling with self similar spl alpha stable processes the fractional levy stable motion model the purpose of the correspondence is the introduction of fractional levy stable motion flsm as a model for signals with long memory and high variability commonly encountered in natural processes we present a concise description of this model from a signal processing viewpoint and its successful application to real world infrared signals for the purpose of resolution enhancement via stochastic interpolation

universal poisson process limits for general random walks abstract this paper considers ensembles of general independent and identically distributed random walks taking the ensemble size to grow infinitely large and also taking the running time of the random walks to grow infinitely large universal poisson process limits are obtained specifically it is established that the positions of general linear random walks converge universally to poisson processes over the real line with uniform and exponential intensities and it is established that the positions of general geometric random walks converge universally to poisson processes over the positive half line with harmonic and power intensities corollaries to these universal convergence results yield the extreme value statistics of gumbel weibull and frechet

studying the sinr process of the typical user in poisson networks using its factorial moment measures based on a stationary poisson point process a wireless network model with random propagation effects shadowing and or fading is considered in order to examine the process formed by the signal to interference plus noise ratio sinr values experienced by a typical user with respect to all the base stations in the down link channel this sinr process is completely characterized by deriving its factorial moment measures which involve numerically tractable explicit integral expressions this novel framework naturally leads to expressions for the k coverage probability including the case of random sinr threshold values considered in multi tier network models while the k coverage probabilities correspond to the marginal distributions of the order statistics of the sinr process a more general relation is presented connecting the factorial moment measures of the sinr process to the joint densities of these order statistics this gives a way for calculating the exact values of the coverage probabilities arising in a general scenario of signal combination and interference cancellation between base stations the presented framework consisting of the mathematical representations of sinr characteristics with respect to the factorial moment measures holds for the whole domain of sinr and is amenable to considerable model extension

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

linear combinations of space time covariance functions and variograms the difference or a linear combination of two space time covariance functions or variograms is not necessarily a valid covariance function or variogram in general there seems no simple condition that makes a linear combination permissible unless its coefficients are non negative the permissibility is investigated in this paper for the linear combination of two spatial or spatio temporal covariance functions or variograms isotropic in space so that we obtain flexible classes of spatial or spatio temporal covariance functions with various properties such as long range dependence and having different signs

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

exact numerical simulation of power law noises many simulations of stochastic processes require colored noises i describe here an exact numerical method to simulate power law noises the method can be extended to more general colored noises and is exact for all time steps even when they are unevenly spaced as may often happen for astronomical data see e g n r lomb astrophys space sci 39 447 1976 the algorithm has a well behaved computational complexity it produces a nearly perfect gaussian noise and its computational efficiency depends on the required degree of noise gaussianity

langevin unification of fractional motions we present a physical langevin based theory explaining the emergence and pervasiveness of the following fractional motions brownian motion l vy motion fractional brownian motion and fractional l vy motion a general form of micro level langevin dynamics with infinitely many degrees of freedom is considered transcending from the micro level to the macro level the infinitely many degrees of freedom collapse to two characteristic exponents and the aforementioned fractional motions emerge universally the exponents categorize the fractional motions and determine their statistical and topological properties the theory establishes a unified langevin bedrock to fractional motions which are widely applied prototypical models of random transport in the sciences

computation of shot noise probability distributions and densities the computation of the cumulative distribution cdf the complementary cdf ccdf and the density of certain shot noise random variables is discussed after subtracting off a few terms that can be computed in closed form what remains can be approximated by a general method for approximating samples of a cdf or ccdf by summing a fourier series whose coefficients are modulated samples of their characteristic function to approximate the density a spline is fit to the cdf samples and then differentiated when the density has corners it is important that the spline have coincident knots at these locations for shot noise densities these locations are easily identified