asymptotic distributions of quasi maximum likelihood estimators for spatial autoregressive models this paper investigates asymptotic properties of the maximum likelihood estimator and the quasi maximum likelihood estimator for the spatial autoregressive model the rates of convergence of those estimators may depend on some general features of the spatial weights matrix of the model it is important to make the distinction with dif ferent spatial scenarios under the scenario that each unit will be influenced by only a few neighboring units the estimators may have n rate of convergence and be asymp totically normal when each unit can be influenced by many neighbors irregularity of the information matrix may occur and various components of the estimators may have different rates of convergence

kernel density estimation for linear processes in this paper we provide a detailed characterization of the asymptotic behavior of kernel density estimators for one sided linear processes the conjecture that asymptotic normality for the kernel density estimator holds under short range dependence is proved under minimal assumptions on bandwidths we also depict the dichotomous and trichotomous phenomena for various choices of bandwidths when the process is long range dependent

on the asymptotic expansion of the empirical process of long memory moving averages let x n i 1 a i e n i where the e i are iid with mean 0 and finite fourth moment and the a i are regularly varying with index where 1 2 1 so that x n has long range dependence this covers an important class of the fractional arima process for r 0 let y n r n n 1 1 j1 jr r s 1 a js e n js y n 0 n 2 n r var y n r and f r the rth derivative of the distribution function of x n the y n r are uncorrelated and are stochastically decreasing in r for any positive integer p 2 1 1 it is shown under mild regularity conditions that with probability 1 n n 1 i x n x p r 0 1 r f r x y n r o n n p uniformly for all x r o 1 2 1 2 p 1 2 this generalizes a host of existing results and provides the vehicle for a number of statistical applications

a korovkin type theory for finite toeplitz operators via matrix algebras preconditioned conjugate gradients pcg are widely and successfully used methods for solving a toeplitz linear system a n vec x vec b 59 9 20 5 34 62 6 10 28 45 44 46 49 frobenius optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the frobenius distance from a n the convergence features of these pcg have been naturally studied by means of the weierstrass jackson theorem 17 36 45 owing to the profound relationship between the spectral features of the matrices a n generated by the fourier coefficients of a continuous function f and the analytical properties of the symbol f itself in this paper we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f on the other hand by modifying the korovkin theorem to study the frobenius optimal preconditioning problem we provide a new and unifying tool for analyzing all frobenius optimal preconditioners in any generic matrix algebra related to trigonometric transforms finally the multilevel case is sketched and discussed by showing that a korovkin type theory also holds in a multivariate sense

absolute equal distribution of families of finite sets let u uin i 1dn n n0 and v vin i 1dn n n0 where u1n u2n udn n v1n v2n vdn n n n0 and limn dn let f be a set of continuous real valued functions on r then u and v are equally distributed with respect to f if i 1dn f uin f vin o dn f f or absolutely equally distributed with respect to f if i 1dn f uin f vin o dn f f we show that these definitions are equivalent if f f c r limx f x andlimx f x exist finite and we give sufficient conditions for u and v to be absolutely equally distributed with respect to f f fisboundedanduniformlycontinuousonr and f f c r fisbounded

existence results and iterative method for solving a nonlinear biharmonic equation of kirchhoff type abstract in this paper for investigating a boundary value problem for a nonlinear biharmonic equation of kirchhoff type we propose an efficient approach by the reduction of the problem to an operator equation for the nonlinear part of the equation the result is that we have established the existence and uniqueness of a solution under some easily verified conditions also we propose an iterative method for finding the solution some examples demonstrate the applicability of the theoretical results and the efficiency of the iterative method

asymptotic expansion of m estimators with long memory errors this paper obtains a higher order asymptotic expansion of a class of m estimators of the one sample location parameter when the errors form a long memory moving average a suitably standardized difference between an m estimator and the sample mean is shown to have a limiting distribution the nature of the limiting distribution depends on the range of the dependence parameter if for example 1 3 then a suitably standardized difference between the sample median and the sample mean converges weakly to a normal distribution provided the common error distribution is symmetric if 0 1 3 then the corresponding limiting distribution is nonnormal this paper thus goes beyond that of beran who observed in the case of long memory gaussian errors that m estimators t n of the one sample location parameter are asymptotically equivalent to the sample mean in the sense that var t n var x n 1 and t n x n o p var x n

central limit theorems for quadratic forms with time domain conditions we establish the central limit theorem for quadratic forms t s 1 n b t s p m n x t x s of the bivariate appell polynomials p m n x t x s under time domain conditions these conditions relate the weights b t and the covariances of the sequences p m n x t x s and x t the time domain approach together with the spectral domain approach developed earlier yields a general set of conditions for central limit theorems

generalized szego theorems and asymptotics of cumulants by graphical methods we obtain some general asymptotics results about a class of deterministic sums called sums with dependent indices which generalize a classical theorem fo szego the above type of sums is encountered when establishing covergence to the gaussian distribution of sums of wick products by the method of cumulants our asymptotic results reduce in this situation the proof of the central limit thoerem to the study of the connectivity of a family of associated graphs

asymptotics of eigenvalues of large symmetric toeplitz matrices with smooth simple loop symbols abstract this paper is devoted to the asymptotic behavior of all eigenvalues of the increasing finite principal sections of an infinite symmetric in general non hermitian toeplitz matrix the symbol of the infinite matrix is supposed to be moderately smooth and to trace out a simple loop in the complex plane the main result describes the asymptotic structure of all eigenvalues the asymptotic expansions constructed are uniform with respect to the location of the eigenvalues in the bulk of the spectrum

the eigenvalue distribution of products of toeplitz matrices clustering and attraction abstract we use a recent result concerning the eigenvalues of a generic non hermitian complex perturbation of a bounded hermitian sequence of matrices to prove that the asymptotic spectrum of the product of toeplitz sequences whose symbols have a real valued essentially bounded product h is described by the function h in the szego way then using mergelyan s theorem we extend the result to the more general case where h belongs to the tilli class the same technique gives us the analogous result for sequences belonging to the algebra generated by toeplitz sequences if the symbols associated with the sequences are bounded and the global symbol h belongs to the tilli class a generalization to the case of multilevel matrix valued symbols and a study of the case of laurent polynomials not necessarily belonging to the tilli class are also given

berry esseen theorems for quadratic forms of gaussian stationary processes letx1 x2 be a gaussian stationary process with spectral densityf and let z j frac 1 sqrt t left sum limits l 1 t j 1 left x l x l j 1 ex l x l j 1 right right j 1 p the present paper gives a berry esseen type theorem for the joint distribution of z1 zp

dynamic spatial panel models networks common shocks and sequential exogeneity this paper considers a class of generalized methods of moments gmm estimators for general dynamic panel models allowing for weakly exogenous covariates and cross sectional dependence due to spatial lags unspecified common shocks and time varying interactive effects we significantly expand the scope of the existing literature by allowing for endogenous time varying spatial weight matrices without imposing explicit structural assumptions on how the weights are formed an important area of application is in social interaction and network models where our specification can accommodate data dependent network formation we consider an exemplary social interaction model and show how identification of the interaction parameters is achieved through a combination of linear and quadratic moment conditions for the general setup we develop an orthogonal forward differencing transformation to aid in the estimation of factor components while maintaining orthogonality of moment conditions this is an important ingredient to a tractable asymptotic distribution of our estimators in general the asymptotic distribution of our estimators is found to be mixed normal due to random norming however the asymptotic distribution of our test statistics is still chi square

properties of nonlinear transformations of fractionally integrated processes this paper shows that the properties of nonlinear transformations of a fractionally integrated process depend strongly on whether the initial series is stationary or not transforming a stationary gaussian i d process with d 0 leads to a long memory process with the same or a smaller long memory parameter depending on the hermite rank of the transformation any nonlinear transformation of an antipersistent gaussian i d process is i 0 for non stationary i d processes every integer power transformation is non stationary and exhibits a deterministic trend in mean and in variance in particular the square of a non stationary gaussian i d process still has long memory with parameter d whereas the square of a stationary gaussian i d process shows less dependence than the initial process simulation results for other transformations are also discussed

some convergence results on quadratic forms for random fields and application to empirical covariances limit theorems are proved for quadratic forms of gaussian random fields in presence of long memory we obtain a non central limit theorem under a minimal integrability condition which allows isotropic and anisotropic models we apply our limit theorems and those of ginovian j contemp math anal 34 2 1 15 to obtain the asymptotic behavior of the empirical covariances of gaussian fields which is a particular example of quadratic forms we show that it is possible to obtain a gaussian limit when the spectral density is not in l 2 therefore the dichotomy observed in dimension d 1 between central and non central limit theorems cannot be stated so easily due to possible anisotropic strong dependence in d 1

singular values and eigenvalues of non hermitian block toeplitz matrices in this paper we are concerned with the analysis of the distribution and localization of the singular values of toeplitz matrices ftn f g generated by a p variate lebesgue integrable matrix valued function f q p c hk q we prove that the union of the essential ranges of the singular values of f is a proper weak cluster for the whole set of the singular values of ftn f g by showing that the number of outliers is strongly depending on the regularity features of the underlying function f in particular if f is continuous or from the krein algebra and p 1 then the cluster is proper other results concerning the extreme spectral behavior of ftn f g second order ergodic formulas and localization of eigenvalues of preconditioned matrices ft 1 n g tn f g are presented some examples of applications to the preconditioning of these results are also discussed c 1999 elsevier science b v all rights reserved msc 15a12 15a18 65f10

existence of nontrivial solutions of an asymptotically linear fourth order elliptic equation abstract in this paper using the mountain pass theorem we give the existence result for nontrivial solutions for a class of asymptotically linear fourth order elliptic equations

asymptotically optimal estimation in misspecified time series models a concept of asymptotically efficient estimation is presented whena misspecified parametric time series model is fitted to a stationary process efficiency of several minimum distance estimates is proved and the behavior ofthe gaussian maximum likelihood estimate is studied furthermore the behaviorof estimates that minimize the h step prediction error is discussed briefly the paper answers to some extent the question what happens when a misspecifiedmodel is fitted to time series data and one acts as if the model were true

an elementary note on asymptotic properties of toeplitz and multilevel toeplitz matrices abstract let a l f l be the fourier coefficients of f l and consider the toeplitz matrices t n f where t n f a i j f i j 1 n thus t n f is generated by the symbol f there are many results on asymptotic properties of t n f as n if r is a positive integer and t n r f a r i j i j 1 n then t n r f t n f r where f r is easily obtained from f hence known results on the asymptotic behavior of toeplitz matrices generated by a symbol can be applied to t n r f although this is elementary to our knowledge it has not been previously exploited we extend this idea to multilevel toeplitz matrices

asymptotic behavior of weighted quadratic and cubic variations of fractional brownian motion this note is devoted to a fine study of the convergence of some weighted quadratic and cubic variations of a fractional brownian motion b with hurst index h in 0 1 2 with the help of malliavin calculus we show that correctly renormalized the weighted quadratic variation of b that we consider converges in l 2 to an explicit limit when h 1 4 in the same spirit we also show that correctly renormalized the weighted cubic variation of b converges in l 2 to an explicit limit when h 1 6

limit theorems for fourier transforms of functionals of gaussian sequences limit theorems with a non gaussian in fact nonstable limiting distribution have been obtained under suitable conditions for partial sums of instantaneous nonlinear functions of stationary gaussian sequences with long range dependence analogous limit theorems are here obtained for finite fourier transforms of instantaneous nonlinear functions of stationary gaussian sequences with long range dependence

central limit theorems for sums of wick products of stationary sequences we show by the method of cumulants that checking whether the central limit theorem for sums of wick powers of a stationary sequence holds can be reduced to the study of an associated graph problem we obtain thus central limit theorems under various integrability conditions on the cumulant spectral functions

the appell interpolation problem a general linear interpolation problem is considered we will call it the appell interpolation problem because the solution can be expressed by a basis of appell polynomials some classical and non classical examples are also considered finally numerical calculations are given

nonlinear system theory another look at dependence based on the nonlinear system theory we introduce previously undescribed dependence measures for stationary causal processes our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems the proposed dependence measures provide a natural framework for a limit theory for stationary processes in particular under conditions with quite simple forms we present limit theorems for partial sums empirical processes and kernel density estimates the conditions are mild and easily verifiable because they are directly related to the data generating mechanisms

the extension of the concept of the generating function to a class of preconditioned toeplitz matrices abstract it is well known that the generating function f l 1 of a class of hermitian toeplitz matrices a n f describes very precisely the spectrum of each matrix of the class recently this theory has been extended in various senses in three directions the non hermitian case the block case and the case of preconditioned toeplitz matrices here by using some results of the first and the second extension we deduce new properties of distribution of the spectrum of preconditioned matrices and we define the concept of generating function for these matrices moreover we discuss the consequences of these results on the analysis of convergence of preconditioned conjugate gradient methods previously defined in the literature finally we perform some numerical experiments which confirm the theoretical idea

finding mixed families of special polynomials associated with appell sequences abstract in this paper certain mixed special polynomial families associated with appell sequences are introduced and their properties are established further operational rules providing connections between these families and known special polynomials are established which are used to derive the identities and results for the members of these new families determinantal definitions of the polynomials associated with appell family are also derived the approach presented is general

pushing the envelope of the test functions in the szego and avram parter theorems the szego and avram parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of hermitian toeplitz matrices and the singular values of arbitrary toeplitz matrices respectively as the matrix dimension goes to infinity the question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous nonnegative and monotonously increasing test functions we show that surprisingly the answer to this question is negative on the other hand we prove the two theorems in a general form which includes all versions known so far

stein s method and exact berry esseen asymptotics for functionals of gaussian fields we show how to detect optimal berry esseen bounds in the normal approximation of functionals of gaussian fields our techniques are based on a combination of malliavin calculus stein s method and the method of moments and cumulants and provide de facto local one term edgeworth expansions the findings of the present paper represent a further refinement of the main results proven in nourdin and peccati probab theory related fields 145 2009 75 118 among several examples we discuss three crucial applications i to toeplitz quadratic functionals of continuous time stationary processes extending results by ginovyan probab theory related fields 100 1994 395 406 and ginovyan and sahakyan probab theory related fields 138 2007 551 579 ii to exploding quadratic functionals of a brownian sheet and iii to a continuous time version of the breuer major clt for functionals of a fractional brownian motion

asymptotic normality of some hermitian forms with complex noisy data hermitian forms with complex random data arise in some areas of physics when one studies the effect of the noise in some frequency interval in this context a central limit theorem is proved for independent gaussian variables in the complex plane the non gaussian case is also studied and the same result holds provided that the fourth order moments are bounded

large deviations for poisson systems of independent random walks we prove large deviation theorems for occupation time functionals of independent random walks started from a poisson field on zd in dimensions 1 and 2 the large deviation tails are larger than exponential exact asymptotics are derived

on the distribution of the singular values of toeplitz matrices in 1920 g szego proved a basic result concerning the distribution of the eigenvalues n j of the toeplitz sections tn f where f l is a real valued function simple examples show that this result cannot hold in the case where f is not real valued in this note we give an extension of this theorem for the singular values of tn f when f f0 r0 with f0 real valued and r0 continuous periodic with period 2 and such that r0 1 in addition we apply the basic theorem of szego to resolve a question of c moler

locally toeplitz sequences spectral properties and applications abstract the notion of locally toeplitz sequence of matrices is introduced which extends the notion of toeplitz sequence of matrices the singular value distribution and in the hermitian case the eigenvalue distribution is completely characterized for locally toeplitz sequences and their sums obtaining weighted szeg formulas which extend well known results due to tyrtyshnikov concerning toeplitz matrices indeed any toeplitz sequence t n f where f is bounded is proved to be a locally toeplitz sequence moreover sufficient conditions are given for the product of two locally toeplitz sequences to be also locally toeplitz by combining these theoretic results we are able to explicitly compute the asymptotic spectral distribution of a large class of matrices arising in the applications including the algebra generated by toeplitz sequences and virtually all matrices resulting from the discretization of a unidimensional differential operator with non constant coefficients finally a large number of examples is discussed

general explicit difference formulas for numerical differentiation through introducing the generalized vandermonde determinant the linear algebraic system of a kind of vandermonde equations is solved analytically by use of the basic properties of this determinant and then we present general explicit finite difference formulas with arbitrary order accuracy for approximating first and higher derivatives which are applicable to unequally or equally spaced data comparing with other finite difference formulas the new explicit difference formulas have some important advantages basic computer algorithms for the new formulas are given and numerical results show that the new explicit difference formulas are quite effective for estimating first and higher derivatives of equally and unequally spaced data

iterative method for solving a nonlinear fourth order boundary value problem in the study of transverse vibrations of a hinged beam there arises a boundary value problem for fourth order ordinary differential equation where a significant difficulty lies in a nonlinear term under integral sign in recent years several authors considered finite approximation of the problem and proposed an iterative method for solving the system of nonlinear equations obtained the essence of the iteration is the simple iteration method for a nonlinear equation although this is not shown in the papers of the authors in this paper we propose a new approach to the solution of the problem which is based on the reduction of it to finding a root of a nonlinear equation in both cases when the explicit form of this equation is found or not the use of the newton or newton type methods generate fast convergent iterative process for the original problem the results of many numerical experiments confirm the efficiency of the proposed approach

limit theorems for toeplitz quadratic functionals of continuous time stationary processes let x t t in mathbb r be a centered real valued stationary gaussian process with spectral density f the paper considers a question concerning asymptotic distribution of toeplitz type quadratic functional qt of the process x t generated by an integrable even function g sufficient conditions in terms of f and g ensuring central limit theorems for standard normalized quadratic functionals qt are obtained extending the results of fox and taqqu prob theory relat fields 74 213 240 1987 avram prob theory relat fields 79 37 45 1988 giraitis and surgailis prob theory relat fields 86 87 104 1990 ginovian and sahakian theory prob appl 49 612 628 2004 for discrete time processes

limit theorems for nonlinear functionals of a stationary gaussian sequence of vectors limit theorems for functions of stationary mean zero gaussian sequences of vectors satisfying long range dependence conditions are considered depending on the rate of decay of the coefficients the limit law can be either gaussian or the law of a multiple ito wiener integral we prove the bootstrap of these limit theorems in the case when the limit is normal a sufficient bracketing condition for these limit theorems to happen uniformly over a class of functions is presented

convergence in distribution of sums of bivariate appell polynomials with long range dependence normalized quadratic forms of moving averages converge to double wiener ito integrals if the summands are sufficiently dependent this result extends to sums of bivariate appell polynomials of arbitrary degree

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

a certain subclass of analytic functions involving operators of fractional calculus making use of certain operators of fractional calculus we introduce a new class t n of functions which are analytic and univalent in the open unit disk u and obtain a necessary and sufficient condition for a function to be in the class t n the various other results presented here for the class t n include the radii of close to convexity starlikeness and convexity and some growth and distortion theorems involving fractional integrals and fractional derivatives some interesting consequences and possible further generalizations of these results are also pointed out

a generalization of the avram parter theorem abstract the first szeg limit theorem gives a remarkable connection between the eigenvalues distribution of a hermitian toeplitz matrix and its symbol this result was extended by avram and parter to the singular values of complex toeplitz matrices the purpose of this note is to extend their results to a larger class of matrices whose entries are equidistributed and have small mean variation we also present applications to kac murdock szeg matrices block toeplitz and locally toeplitz matrices

k terms recurrence relations and polynomial variance functions of the k th degree abstract the aim of this paper is to investigate the recurrence relations obtained for some polynomials associated with natural exponential families via their variance functions in the case of polynomial variance functions of the k th degree we obtain a recurrence relation with k terms a few examples are indicated and the orthogonality of such polynomials is characterized this characterization is due to pommeret test 5 1996 77 111 moreover some of the recent results related to lie algebras levy processes and martingales are reviewed in a more general setting

influence of matrix operations on the distribution of eigenvalues and singular values of toeplitz matrices abstract suppose some toeplitz matrix families a n are given generated by the fourier expansions for and a new family a n is constructed from a n via basic matrix operations theorems are proved that describe the singular value distribution for a n in the terms of as well as the eigenvalue distribution for h a n a n a n 2 and k a n a n a n 2i in particular if l and only multiplication is used then we show the singular values of a n are distributed as x where x x at the same time the eigenvalues of h a n are distributed as re x while those of k a n are distributed as im x the extension to multilevel toeplitz matrices is also suggested finally an application to circulant preconditioning is discussed

central limit theorems for sequences of multiple stochastic integrals we characterize the convergence in distribution to a standard normal law for a sequence of multiple stochastic integrals of a fixed order with variance converging to 1 some applications are given in particular to study the limiting behavior of quadratic functionals of gaussian processes

circulant preconditioners with unbounded inverses abstract the eigenvalue and singular value distributions for matrices s 1 n a n and c 1 n a n are examined where a n s n and c n are toeplitz matrices simple circulants and optimal circulants generated by the fourier expansion of some function f recently it has been proven that a cluster at 1 exists whenever f is from the wiener class and strictly positive both restrictions are now weakened a proof is given for the case when f may take the zero value and hence the circulants are to have unbounded inverses the main requirements on f are that it belong to l 2 and be in some sense sparsely vanishing specifically if f is nonnegative and circulants s n or c n are positive definite then the eigenvalues of s 1 n a n or c 1 n a n are clustered at 1 if f is complex valued and s n or c n are nonsingular then the singular values of s 1 n a n or c 1 n a n are clustered at 1 as well also proposed and studied are the improved circulants it is shown that improved simple circulants can be much more advantageous than optimal circulants this depends crucially on the smoothness properties of f further clustering on theorems are given that pertain to multilevel toeplitz matrices preconditioned by multilevel simple and optimal circulants

clt and other limit theorems for functionals of gaussian processes conditions for the clt for non linear functionals of stationary gaussian sequences are discussed with special references to the borderline between the clt and the non clt examples of the non clt for such functionals with the norming factor sqrt n are given

generalized locally toeplitz sequences spectral analysis and applications to discretized partial differential equations abstract starting from the finite difference discretization of an elliptic second order pde as i j 1 d x i a i j x x j u x b x over a bounded domain we introduce the notion of generalized locally toeplitz sequence of matrices the singular value distribution the eigenvalue distribution in the hermitian case is studied and characterized for generalized locally toeplitz sequences in terms of weighted multidimensional szeg formulas this extends preceding results attributed to tilli which concern the unilevel case the application of this theoretic analysis to the numerical solution of pdes is finally discussed

the glt class as a generalized fourier analysis and applications recently the class of generalized locally toeplitz glt sequences has been introduced as a generalization both of classical toeplitz sequences and of variable coefficient differential operators and for every sequence of the class it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function the symbol that can be easily identified this generalizes the notion of a symbol for differential operators discrete and continuous or for toeplitz sequences for which it is identified through the fourier coefficients and is related to the classical fourier analysis the glt class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations and products in this paper we prove that the considered class is closed under inversion as well when the sequence which is inverted shows a sparsely vanishing symbol sparsely vanishing symbol a symbol which vanishes at most in a set of zero lebesgue measure furthermore we show that the glt class virtually includes any finite difference or finite element discretization of pdes and based on this we demonstrate that our results on glt sequences can be used in a pde setting in various directions 1 as a generalized fourier analysis for the study of iterative and semi iterative methods when dealing with variable coefficients non rectangular domains non uniform gridding or triangulations 2 in order to provide a tool for the stability analysis of pde numerical schemes e g a necessary von neumann criterium for variable coefficient systems of pdes is obtained uniformly with respect to the boundary conditions 3 for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case etc the final part of the paper deals indeed with problems 1 3 and other possible directions in which the glt analysis can be conveniently employed

spectra of multilevel toeplitz matrices advanced theory via simple matrix relationships we consider the eigenvalue and singular value distributions for m level toeplitz matrices generated by a complex valued periodic function of m real variables we show that familiar formulations for l due to szeg and others can be preserved so long as f l1 and what is more with g weyl s definitions just a bit changed in contrast to other approaches the one we follow is based on simple matrix relationships

stein s method on wiener chaos we combine malliavin calculus with stein s method in order to derive explicit bounds in the gaussian and gamma approximations of random variables in a fixed wiener chaos of a general gaussian process our approach generalizes refines and unifies the central and non central limit theorems for multiple wiener ito integrals recently proved in several papers from 2005 to 2007 by nourdin nualart ortiz latorre peccati and tudor we apply our techniques to prove berry esseen bounds in the breuer major clt for subordinated functionals of fractional brownian motion by using the well known mehler s formula for ornstein uhlenbeck semigroups we also recover a technical result recently proved by chatterjee concerning the gaussian approximation of functionals of finite dimensional gaussian vectors

existence of nontrivial solutions for fourth order asymptotically linear elliptic equations abstract in this paper we discuss the following fourth order semilinear elliptic problem 2 u c u f x u x u u 0 x where f x t is asymptotically linear with respect to t at infinity is a smooth bounded domain in r n and n 4 in this case f x t does not satisfy the following ambrosetti rabinowitz type condition see ambrosetti and rabinowitz 1973 9 for short which is called the ar condition that is for some 0 and m 0 0 f x t 0 t f x s d s 1 2 f x t t uniformly a e x and t m which is important in applying the mountain pass theorem by a variant version of the mountain pass theorem we obtain the existence of nontrivial solutions to the above problem under suitable assumptions of f x t which generalizes and improves the results in liu and wang 2007 12 and an and liu 2008 13

on bilinear forms in gaussian random variables and toeplitz matrices we improve a result of szeg on the asymptotic behaviour of the trace of products of toeplitz matrices

topological foundations of an asymptotic approximation theory for sequences of matrices with increasing size abstract sequences of matrices with increasing size arise in several contexts including the discretization of integral and differential equations an asymptotic approximation theory for this kind of sequences has recently been developed with the aim of providing tools for computing their asymptotic singular value and eigenvalue distribution the basis of this theory is the notion of approximating classes of sequences a c s which is also fundamental to the theory of generalized locally toeplitz sequences and hence to the spectral analysis of pde discretization matrices in this paper we show that the a c s notion is a convergence notion induced by a pseudometrizable topology we also identify a corresponding pseudometric and we study some of its properties it turns out that there exists a strong connection between the a c s topology and the topology of convergence in measure

singular values of cauchy toeplitz matrices abstract the behavior of singular values of matrices a n 1 i j g n i j 1 with n is investigated for any real g which is not integer it is proved that the singular values are clustered at sin g which is their upper boundary the only o n singular values are those which lie outside a given e neighborhood of the clustering point o n n 0 as n o n o ln 2 n holds if g 1 2 also proved is that the minimum singular values of a n g tend to zero provided that g 1 2

construction of stationary self similar generalized fields by random wavelet expansion random wavelet expansion is introduced in the study of stationary self similar generalized random fields it is motivated by a model of natural images in which 2d views of objects are randomly scaled and translated because the objects are randomly distributed in the 3d space it is demonstrated that any stationary self similar random field defined on the dual space of a schwartz space of smooth rapidly decreasing functions has a random wavelet expansion representation to explicitly construct stationary self similar random fields random wavelet expansion representations incorporating random functionals of the following three types are considered 1 a multiple stochastic integral over the product domain of scale and translate 2 an iterated one first integrating over the scale domain and 3 an iterated one first integrating over the translate domain we show that random wavelet expansion gives rise to a variety of stationary self similar random fields including such well known processes as the linear fractional stable motions

a note on the spectral distribution of symmetrized toeplitz sequences abstract the singular value and spectral distribution of toeplitz matrix sequences with lebesgue integrable generating functions is well studied early results were provided in the classical szeg theorem and the avram parter theorem in which the singular value symbol coincides with the generating function more general versions of the theorem were later proved by zamarashkin and tyrtyshnikov and tilli considering real nonsymmetric toeplitz matrix sequences we first symmetrize them via a simple permutation matrix and then we show that the singular value and spectral distribution of the symmetrized matrix sequence can be obtained analytically by using the notion of approximating class of sequences in particular under the assumption that the symbol is sparsely vanishing we show that roughly half of the eigenvalues of the symmetrized toeplitz matrix i e a hankel matrix are negative positive for sufficiently large dimension i e the matrix sequence is symmetric asymptotically indefinite

a central limit theorem for the kpz equation we consider the kpz equation in one space dimension driven by a stationary centred space time random field which is sufficiently integrable and mixing but not necessarily gaussian we show that in the weakly asymmetric regime the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the hopf cole solution to the kpz equation driven by space time gaussian white noise while the limiting process depends only on the integrated variance of the driving field the diverging constants appearing in the definition of the reference frame also depend on higher order moments

toeplitz eigenvalues for radon measures abstractit is well known that for toeplitz matrices generated by a suf ciently smooth real valuedsymbol the eigenvalues behave asymptotically as the values of the symbol on uniform mesheswhile the singular values even for complex valued functions do as those values in modulus these facts are expressed analytically by the szego and szego like formulas and as is provedrecently the smoothness assumptions are as mild as those of l 1 in this paper it is shownthat the szego like formulas hold true even for toeplitz matrices generated by the so calledradon measures 2002 elsevier science inc all rights reserved ams classi cation 15a12 65f10 65f15 65t10keywords toeplitz matrices eigenvalues singular values szego formulas radon measures 1 introductionwe consider a sequence of toeplitz matrices cf 3 a n a kl a kl a k l 0 k l n 1 1 constructed from the coef cients of a formal fourier series f x k a k e i kx 2 supported by the russian fund of basic research under grant 99 01 00017 and completed while the rst author was a guest professor at the university of saarland germany under support of daad

a certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients recently the authors made use of a certain operator of fractional derivatives in order to introduce and initiate a systematic study of a novel subclass tp of analytic and p valent functions with negative coefficients in this sequel to the aforementioned work they prove a number of closure and inclusion theorems and determine the radii of p valently close to convexity starlikeness and convexity for the class tp they also obtain a class preserving integral operator of the form f z j p z pz z0t 1 t dt p for the class studied here

asymptotic distributions of the sample mean autocovariances and autocorrelations of long memory time series we derive the asymptotic distributions of the sample mean autocovariances and autocorrelations for a time series whose autocovariance function has the powerlaw decay 0 o 1 as the results differ in important respects from the corresponding results for short memory processes whose autocovariance functions are absolutely summable for long memory processes the variances of the sample mean and of the sample autocovariances and autocorrelations for 0 12 are not of asymptotic order n 1 when 0 12 the asymptotic distributions of the sample autocovariances and autocorrelations are not normal

band toeplitz preconditioners for non symmetric real toeplitz systems by preconditioned gmres method abstract in this paper we study n n real non symmetric ill conditioned toeplitz systems of the form t n f x b the corresponding generating function is a complex valued one of the form f f 1 i f 2 where i 1 is known a priori and has roots we note that f 1 is a 2 periodic even function while f 2 is a 2 periodic odd one in order to solve the above system efficiently we use the preconditioned generalized minimal residual pgmres method and the preconditioned conjugate gradient method applied to the normal equations pcgn we present a specific preconditioning technique that combines elimination of the roots of f and best uniform approximation or interpolation by trigonometric polynomials the proposed preconditioner is a band toeplitz matrix t n p generated by the trigonometric polynomial p g q the trigonometric polynomial g is an appropriate complex polynomial having the same zeros as f while q is the trigonometric polynomial derived by the uniform approximation or interpolation of the function f g using t n p as a preconditioner we achieve a good clustering of the singular values of the preconditioned matrix in a small interval around 1 as well as a good clustering of the eigenvalues in a small domain in the right half plane of the complex plane far from zero moreover we describe on how the presented preconditioning technique can be extended to the two level toeplitz systems finally we show by various numerical experiments that the proposed preconditioning technique can solve a toeplitz system of the above form in a small number of iterations

on toeplitz type quadratic functionals of stationary gaussian processes a central limit theorem for toeplitz type quadratic functionals of a stationary gaussian processx t t is proved generalizing the result of avram 1 for discrete time processes the result is applied to the problem of nonparametric estimation of linear functionals of an unknown spectral density function we give some upper bounds for the minimax mean square risk of the nonparametric estimators similar to those by ibragimov and has minskii 12 for a probability density function

on the asymptotic equivalence of circulant and toeplitz matrices any sequence of uniformly bounded n times n hermitian toeplitz matrices h n is asymptotically equivalent to a certain sequence of n times n circulant matrices c n derived from the toeplitz matrices in the sense that left h n c n right f o sqrt n as n rightarrow infty this implies that certain collective behaviors of the eigenvalues of each toeplitz matrix are reflected in those of the corresponding circulant matrix and supports the utilization of the computationally efficient fast fourier transform instead of the karhunen loeve transform in applications like coding and filtering in this paper we study the asymptotic performance of the individual eigenvalue estimates we show that the asymptotic equivalence of the circulant and toeplitz matrices implies the individual asymptotic convergence of the eigenvalues for certain types of toeplitz matrices we also show that these estimates asymptotically approximate the largest and smallest eigenvalues for more general classes of toeplitz matrices

on sign changing solution for a fourth order asymptotically linear elliptic problem in this paper we deal with a fourth order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity by using the variational method we obtain an existence result of sign changing solutions as well as positive and negative solutions

limit theorems for functionals of moving averages let x n i 1 a i e n 1 where the e i are i i d with mean 0 and finite second moment and the a i are either summable or regularly varying with index 1 1 2 the sequence x n has short memory in the former case and long memory in the latter for a large class of functions k a new approach is proposed to develop both central n rate and noncentral non n rate limit theorems for s n n 1 n k x n ek x n specifically we show that in the short memory case the central limit theorem holds for s n and in the long memory case s n can be decomposed into two asymptotically uncorrelated parts that follow a central limit and a non central limit theorem respectively further we write the noncentral part as an expansion of uncorrelated components that follow noncentral limit theorems connections with the usual hermite expansion in the gaussian setting are also explored

limit theorems for non linear functionals of gaussian sequences we prove limit theorems for sums of non linear functionals of gaussian sequences in certain cases we obtain a non gaussian limit with a norming factor n 0 1 2 the class of functionals we are investigating is a natural enlargement of the class investigated by m rosenblatt in 7 we prove our results by refining the method of the paper 3

iterative method for solving the neumann boundary value problem for biharmonic type equation the solution of boundary value problems bvp for fourth order differential equations by their reduction to bvp for second order equations with the aim to use the achievements for the latter ones attracts attention from many researchers in this paper using the technique developed by ourselves in recent works we construct iterative method for the neumann bvp for biharmonic type equation the convergence rate of the method is proved and some numerical experiments are performed for testing it in dependence on the choice of an iterative parameter

power variation for a class of stationary increments levy driven moving averages in this paper we present some new limit theorems for power variation of kkth order increments of stationary increments levy driven moving averages in the infill asymptotic setting where the sampling frequency converges to zero while the time span remains fixed the asymptotic theory gives novel results which partially have no counterpart in the theory of discrete moving averages more specifically we show that the first order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k 1k 1 the considered power p 0p 0 the blumenthal getoor index 0 2 0 2 of the driving pure jump levy process ll and the behaviour of the kernel function gg at 00 determined by the power first order asymptotic theory essentially comprises three cases stable convergence towards a certain infinitely divisible distribution an ergodic type limit theorem and convergence in probability towards an integrated random process we also prove a second order limit theorem connected to the ergodic type result when the driving levy process ll is a symmetric stable process we obtain two different limits a central limit theorem and convergence in distribution towards a k k stable totally right skewed random variable

on the asymptotic distribution of the moran i test statistic with applications by far the most popular test for spatial correlation is the one based on moran s 1950 i test statistic despite this the available results in the literature concerning the large sample distribution of this statistic are limited and have been derived under assumptions that do not cover many applications of interest in this paper we 3rst give a general result concerning the large sample distribution of moran i type test statistics we then apply this result to derive the large sample distribution of the moran i test statistic for a variety of important models in order to establishth ese results we also give a new central limit theorem for linear quadratic forms 2001 elsevier science s a all rights reserved jel classi cation c12 c21

an application of fractional calculus to a new class of multivalent functions with negative coefficients abstract motivated by some earlier works of chen et al cf 1 2 dealing with various applications of the operators of fractional calculus in analytic function theory the authors introduce and study rather systematically a certain subclass of analytic and p valent functions with negative coefficients this subclass is defined by using a familiar fractional derivative operator coefficient estimates growth and distortion theorems and many other interesting and useful properties and characteristics of this class of analytic and p valent functions are obtained some of these properties involve for example linear combinations and modified hadamard products or convolution of functions belonging to the class introduced here