space time music consistent signal subspace estimation for wideband sensor arrays wideband direction of arrival doa estimation with sensor arrays is an essential task in sonar radar acoustics biomedical and multimedia applications many state of the art wideband doa estimators coherently process frequency binned array outputs by approximate maximum likelihood ml weighted subspace fitting or focusing techniques this paper shows that bin signals obtained by filter bank approaches do not obey the finite rank narrow band array model because spectral leakage and changes of the array response with frequency within the bin create ghost sources dependent on the particular realization of the source process therefore existing doa estimators based on binning are not consistent even when the array response is perfectly known in this paper under a more realistic array model which still has finite rank under a space time formulation signal subspaces at arbitrary frequencies can be consistently recovered under mild conditions by applying space time music type st music estimators to the dominant eigenvectors of the wideband space time sensor cross correlation matrix a novel consistent ml based st music subspace estimate is developed to estimate the number of sources active at each frequency by information theoretic criteria empirical st music subspaces are fed to any subspace fitting doa estimator at single or multiple frequencies simulations confirm that this approach allows better performance over binning approaches at sufficiently high signal to noise ratio when model mismatches exceed the noise floor

limiting spectral distribution of gram matrices associated with functionals of β mixing processes we give asymptotic spectral results for gram matrices of the form n 1 x n x t n where the entries of x n are dependent across both rows and columns and that are functionals of absolutely regular sequences and have only finite second moments we derive under mild dependence conditions in addition to an arithmetical decay condition on the mixing coefficients an integral equation of the stieltjes transform of the limiting spectral distribution of n 1 x n x t n in terms of the spectral density of the underlying process applications to examples of positive recurrent markov chains and dynamical systems are also given

markov chain monte carlo 10 years and still running although markov chain monte carlo mcmc techniques can be traced back to the early 1950s and are almost contemporary with independent monte carlo methods the use of these techniques has only really taken off in the past 10 years at least in mainstream statistics with the work of gelfand and smith 1990 and the rediscovery of metropolis s and hastings s papers the impact on the discipline is deep and durable because these methods have opened new horizons in the scale of the problems that one can deal with thus enhancing the position of statistics in most applied fields the mcmc revolution has in particular boosted bayesian statistics to new heights by providing a virtually universal tool for dealing with integration and optimization problems this can be seen in for instance the explosion of papers dealing with complex models hierarchical modelings nonparametric bayesian estimation and spatial statistics this trend has also created new synergies with mathematicians and probabilists as well as econometricians engineers ecologists astronomers and others for theoretical requests and practical implications of mcmc techniques it is impossible to provide here a complete perspective on mcmc methods and applications for this as well as for other references we refer the reader to the books by gilks richardson and spiegelhalter 1995 and robert and casella 1999 here we focus on four particular aspects of mcmc algorithms theoretical foundations practical implementation bayesian inference and prospects the web provides many relevant sites including the mcmc preprint server http www statslab cam ac uk r mcmc which offers a most useful window to the current research on mcmc wilson s site on perfect simulation http dimacs rutgers edu dbwilson exact h which keeps track of papers and applets on this topic and a site on convergence control techniques http www ensae fr crest statistique robert mcdiag even more recent developments allow for on line computations through applets like breyer s http www maths lancs ac uk breyer java html which impressively illustrates the behavior of different samplers for simple problems note also that the bugs and coda software are freely available at http www mrc bsu cam ac uk bu

illustration of bayesian inference in normal data models using gibbs sampling abstract the use of the gibbs sampler as a method for calculating bayesian marginal posterior and predictive densities is reviewed and illustrated with a range of normal data models including variance components unordered and ordered means hierarchical growth curves and missing data in a crossover trial in all cases the approach is straightforward to specify distributionally and to implement computationally with output readily adapted for required inference summaries

on slow fading mimo systems with nonseparable correlation in a frequency selective slow fading channel in a multiple input multiple output mimo system the channel matrix is of the form of a block matrix a method is proposed to calculate the limit of the eigenvalue distribution of block matrices if the size of the blocks tends to infinity asymptotic eigenvalue distribution of is also calculated where the entries of are jointly gaussian with a correlation of the form where is fixed and does not increase with the size of the matrix an operator valued free probability approach is used to achieve this goal using this method a system of equations is derived which can be solved numerically to compute the desired eigenvalue distribution

freely independent random variables with non atomic distributions 2015 american mathematical society we examine the distributions of non commutative polynomials of non atomic freely independent random variables in particular we obtain an analogue of the strong atiyah conjecture for free groups thus proving that the measure of each atom of any n n matricial polynomial of nonatomic freely independent random variables is an integer multiple of n 1 in addition we show that the cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real analytic except at a finite number of points

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

antithetic acceleration of monte carlo integration in bayesian inference abstract it is proposed to sample antithetically rather than randomly from the posterior density in bayesian inference using monte carlo integration conditions are established under which the number of replications required with antithetic sampling relative to the number required with random sampling is inversely proportional to sample size as sample size increases the result is illustrated in an experiment using a bivariate vector autoregression

local spectral statistics of gaussian matrices with correlated entries we prove optimal local law bulk universality and non trivial decay for the off diagonal elements of the resolvent for a class of translation invariant gaussian random matrix ensembles with correlated entries

eigenvector dynamics general theory and some applications we propose a general framework to study the stability of the subspace spanned by p consecutive eigenvectors of a generic symmetric matrix h0 when a small perturbation is added this problem is relevant in various contexts including quantum dissipation h0 is then the hamiltonian and financial risk control in which case h0 is the assets return covariance matrix we argue that the problem can be formulated in terms of the singular values of an overlap matrix which allows one to define an overlap distance we specialize our results for the case of a gaussian orthogonal h0 for which the full spectrum of singular values can be explicitly computed we also consider the case when h0 is a covariance matrix and illustrate the usefulness of our results using financial data the special case where the top eigenvalue is much larger than all the other ones can be investigated in full detail in particular the dynamics of the angle made by the top eigenvector and its true direction defines an interesting class of random processes

fluctuations of eigenvalues and second order poincare inequalities linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems the proofs of such results are usually rather difficult involving hard computations specific to the model in question in this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments in the process we introduce a notion of second order poincare inequalities just as ordinary poincare inequalities give variance bounds second order poincare inequalities give central limit theorems the proof of the main result employs stein s method of normal approximation a number of examples are worked out some of which are new one of the new results is a clt for the spectrum of gaussian toeplitz matrices

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

a law of large numbers for finite range dependent random matrices we consider random hermitian matrices in which distant above diagonal entries are independent but nearby entries may be correlated we find the limit of the empirical distribution of eigenvalues by combinatorial methods we also prove that the limit has an algebraic stieltjes transform by an argument based on dimension theory of noetherian local rings 2008 wiley periodicals inc

a domain decomposition method for pseudo spectral electromagnetic simulations of plasmas abstract pseudo spectral electromagnetic solvers i e representing the fields in fourier space have extraordinary precision in particular haber et al presented in 1973 a pseudo spectral solver that integrates analytically the solution over a finite time step under the usual assumption that the source is constant over that time step yet pseudo spectral solvers have not been widely used due in part to the difficulty for efficient parallelization owing to global communications associated with global ffts on the entire computational domains a method for the parallelization of electromagnetic pseudo spectral solvers is proposed and tested on single electromagnetic pulses and on particle in cell simulations of the wakefield formation in a laser plasma accelerator the method takes advantage of the properties of the discrete fourier transform the linearity of maxwell s equations and the finite speed of light for limiting the communications of data within guard regions between neighboring computational domains although this requires a small approximation test results show that no significant error is made on the test cases that have been presented the proposed method opens the way to solvers combining the favorable parallel scaling of standard finite difference methods with the accuracy advantages of pseudo spectral methods

bayesian inference of survival probabilities under stochastic ordering constraints abstract in the statistical analysis of survival data arising from two populations it often happens that the analyst knows a priori that the life lengths in one population are stochastically shorter than those in the other nevertheless survival probability estimates if determined separately from the corresponding samples may not be consistent with this prior assumption because of inherent statistical variability in the observations this problem has been considered in a number of papers during the past decade by adopting a generalized maximum likelihood approach our approach is bayesian and in essence nonparametric the a priori assumption regarding stochastic ordering is formulated naturally in terms of a joint prior distribution defined for pairs of survival functions nonparametric specification of the model based on hazard rates and using a few hyperparameters allows for sufficient flexibility in practical applications the numerical computations are based on a coupled version of the m

on the norm and eigenvalue distribution of large random matrices we study the eigenvalue distribution of n n symmetric random matrices h n x y n 1 2 h x y x y 1 n where h x y x y are gaussian weakly dependent random variables we prove that the normalized eigenvalue counting function of h n converges with probability 1 to a nonrandom function as n we derive an equation for the stieltjes transform of the measure d and show that the latter has a compact support we find the upper bound for lim sup n h n and study asymptotically the case when there are no eigenvalues of h n outside of when n

a convergent born series for solving the inhomogeneous helmholtz equation in arbitrarily large media we present a fast method for numerically solving the inhomogeneous helmholtz equation our iterative method is based on the born series which we modified to achieve convergence for scattering media of arbitrary size and scattering strength compared to pseudospectral time domain simulations our modified born approach is two orders of magnitude faster and nine orders of magnitude more accurate in benchmark tests in 1 2 and 3 dimensional systems

universality for general wigner type matrices we consider the local eigenvalue distribution of large self adjoint n times n random matrices mathbf h mathbf h with centered independent entries in contrast to previous works the matrix of variances s i j mathbbm e h i j 2 is not assumed to be stochastic hence the density of states is not the wigner semicircle law its possible shapes are described in the companion paper ajanki et al in quadratic vector equations on the complex upper half plane arxiv 1506 05095 we show that as n grows the resolvent mathbf g z mathbf h z 1 converges to a diagonal matrix mathrm diag mathbf m z where mathbf m z m 1 z dots m n z solves the vector equation 1 m i z z sum j s i j m j z that has been analyzed in ajanki et al quadratic vector equations on the complex upper half plane arxiv 1506 05095 we prove a local law down to the smallest spectral resolution scale and bulk universality for both real symmetric and complex hermitian symmetry classes

a clt for a band matrix model a law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on or above diagonal entries are independent but neither necessarily identically distributed nor necessarily all of the same variance the derivation is based on systematic combinatorial enumeration study of generating functions and concentration inequalities of the poincare type special cases treated with an explicit evaluation of limiting variances are generalized wigner and wishart matrices

antithetic coupling of two gibbs sampler chains two coupled gibbs sampler chains both with invariant probability density are run in parallel so that the chains are negatively correlated we define an asymptotically unbiased estimator of the expectation e f x which achieves significant variance reduction with respect to the usual gibbs sampler at comparable computational cost the variance of the estimator based on the new algorithm is always smaller than the variance of a single gibbs sampler chain if is attractive and f is monotone nondecreasing in all components of x for nonattractive targets our results are not complete the new antithetic algorithm outperforms the standard gibbs sampler when is a multivariate normal density or the ising model more generally nonrigorous arguments and numerical experiments support the usefulness of the antithetically coupled gibbs samplers also for other nonattractive models in our experiments the variance is reduced to at least a third and the efficiency also improves significantly

a generalization of the lindeberg principle we generalize lindeberg s proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables the result is applied to get a similar theorem for smooth functions of exchangeable random variables this theorem allows us to identify for the first time the limiting spectral distributions of wigner matrices with exchangeable entries

a particle in cell method for the simulation of plasmas based on an unconditionally stable field solver abstract we propose a new particle in cell pic method for the simulation of plasmas based on a recently developed unconditionally stable solver for the wave equation this method is not subject to a cfl restriction limiting the ratio of the time step size to the spatial step size typical of explicit methods while maintaining computational cost and code complexity comparable to such explicit schemes we describe the implementation in one and two dimensions for both electrostatic and electromagnetic cases and present the results of several standard test problems showing good agreement with theory with time step sizes much larger than allowed by typical cfl restrictions

limiting spectral distribution of a new random matrix model with dependence across rows and columns abstract we introduce a random matrix model where the entries are dependent across both rows and columns more precisely we investigate matrices of the form x x i 1 n t it r p n derived from a linear process x t j c j z t j where the z t are independent random variables with bounded fourth moments we show that when both p and n tend to infinity such that the ratio p n converges to a finite positive limit y the empirical spectral distribution of p 1 xx t converges almost surely to a deterministic measure this limiting measure which depends on y and the spectral density of the linear process x t is characterized by an integral equation for its stieltjes transform the matrix p 1 xx t can be interpreted as an approximation to the sample covariance matrix of a high dimensional process whose components are independent copies of x t

covariance structure and convergence rate of the gibbs sampler with various scans this paper presents results on covariance structure and convergence for the gibbs sampler with both systematic and random scans it is shown that under conditions that guarantee the compactness of the markov forward operator and irreducibility of the corresponding chain the gibbs sampling scheme converges geometrically in terms of pearson 2 distance in particular for the random scan the autocovariance can be expressed as variances of iterative conditional expectations as a consequence the autocorrelations are all positive and decrease monotonically

stability of the matrix dyson equation and random matrices with correlations we consider real symmetric or complex hermitian random matrices with correlated entries we prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum the correlations have fast decay but are otherwise of general form the key novelty is the detailed stability analysis of the corresponding matrix valued dyson equation whose solution is the deterministic limit of the resolvent

covariance structure of the gibbs sampler with applications to the comparisons of estimators and augmentation schemes summary we study the covariance structure of a markov chain generated by the gibbs sampler with emphasis on data augmentation when applied to a bayesian missing data problem the gibbs sampler produces two natural approximations for the posterior distribution of the parameter vector the empirical distribution based on the sampled values of the parameter vector and a mixture of complete data posteriors we prove that rao blackwellization causes a one lag delay for the autocovariances among dependent samples obtained from data augmentation and consequently the mixture approximation produces estimates with smaller variances than the empirical approximation the covariance structure results are used to compare different augmentation schemes it is shown that collapsing and grouping random components in a gibbs sampler with two or three components usually result in more efficient sampling schemes

asymptotic properties of large random matrices with independent entries we study the normalized trace gn z n 1 tr h zi 1 of the resolvent of n n real symmetric matrices h 1 jk wjk n j k 1n assuming that their entries are independent but not necessarily identically distributed random variables we develop a rigorous method of asymptotic analysis of moments of gn z for iz 0 where 0 is determined by the second moment of wjk by using this method we find the asymptotic form of the expectation e gn z and of the connected correlator e gn z1 gn z2 e gn z1 e gn z2 we also prove that the centralized trace ngn z e ngn z has the gaussian distribution in the limit n based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles

multiprocess parallel antithetic coupling for backward and forward markov chain monte carlo antithetic coupling is a general stratification strategy for reducing monte carlo variance without increasing the simulation size the use of the antithetic principle in the monte carlo literature typically employs two strata via antithetic quantile coupling we demonstrate here that further stratification obtained by using k 2 e g k 3 10 antithetically coupled variates can offer substantial additional gain in monte carlo efficiency in terms of both variance and bias the reason for reduced bias is that antithetically coupled chains can provide a more dispersed search of the state space than multiple independent chains the emerging area of perfect simulation provides a perfect setting for implementing the k process parallel antithetic coupling for mcmc because without antithetic coupling this class of methods delivers genuine independent draws furthermore antithetic backward coupling provides a very convenient theoretical tool for investigating antithetic forward coupling however the generation of k 2 antithetic variates that are negatively associated that is they preserve negative correlation under monotone transformations and extremely antithetic that is they are as negatively correlated as possible is more complicated compared to the case with k 2 in this paper we establish a theoretical framework for investigating such issues among the generating methods that we compare latin hypercube sampling and its iterative extension appear to be generalpurpose choices making another direct link between monte carlo and quasi monte carlo