berry esseen theorems under weak dependence let xk k z be a stationary sequence given p 2 3 moments and a mild weak dependence condition we show a berry esseen theorem with optimal rate np 2 1 for p 4 we also show a convergence rate of n1 2 in lq norm where q 1 up to logn factors we also obtain nonuniform rates for any p 2 this leads to new optimal results for many linear and nonlinear processes from the time series literature but also includes examples from dynamical system theory the proofs are based on a hybrid method of characteristic functions coupling and conditioning arguments and ideal metrics

common message broadcast channels with feedback in the nonasymptotic regime full feedback we investigate the maximum coding rate for a given average blocklength and error probability over a k user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable length stop feedback codes for the point to point case polyanskiy et al 2011 demonstrated that variable length coding combined with stop feedback significantly increases the speed of convergence of the maximum coding rate to capacity this speed up manifests itself in the absence of a square root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths i e zero dispersion in this paper we present nonasymptotic achievability and converse bounds on the maximum coding rate of the common message k user discrete memoryless broadcast channel which strengthen and generalize the ones reported in trillingsgaard et al 2015 for the two user case an asymptotic analysis of these bounds reveals that zero dispersion cannot be achieved for certain common message broadcast channels e g the binary symmetric broadcast channel furthermore we identify conditions under which our converse and achievability bounds are tight up to the second order through numerical evaluations we illustrate that our second order expansions approximate accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point to point case

on the bootstrap for moran s i test for spatial dependence this paper is concerned with the use of the bootstrap for statistics in spatial econometric models with a focus on the test statistic for moran s i test for spatial dependence we show that for many statistics in spatial econometric models the bootstrap can be studied based on linear quadratic lq forms of disturbances by proving the uniform convergence of the cumulative distribution function for lq forms to that of a normal distribution we show that the bootstrap is generally consistent for test statistics that can be approximated by lq forms including moran s i possible asymptotic refinements of the bootstrap are most commonly studied using edgeworth expansions for spatial econometric models we may establish asymptotic refinements of the bootstrap based on asymptotic expansions of lq forms when the disturbances are normal we prove the existence of the usual edgeworth expansions for lq forms when the disturbances are not normal we establish an asymptotic expansion of lq forms based on martingales these results are applied to show the second order correctness of the bootstrap for moran s i test

generalized gamma approximation with rates for urns walks and trees we study a new class of time inhomogeneous polya type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters a class which includes the rayleigh half normal and gamma distributions our main tool is stein s method combined with characterizing the generalized gamma limiting distributions as fixed points of distributional transformations related to the equilibrium distributional transformation from renewal theory we identify special cases of these urn models in recursive constructions of random walk paths and trees yielding rates of convergence for local time and height statistics of simple random walk paths as well as for the size of random subtrees of uniformly random binary and plane trees

the precise asymptotic behavior of parameter estimators in ornstein uhlenbeck process abstract in the present paper we study the asymptotic behavior for estimator of the drift parameter in an ornstein uhlenbeck process the l r convergence rate and the precise asymptotics in the law of iterated logarithm and in the law of logarithm for the estimator are obtained moreover we also get the complete moment convergence of this estimator the main method of this paper is the deviation inequality for the quadratic functional

products of many large random matrices and gradients in deep neural networks we study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity our main theorem is that the logarithm of the ell 2 norm of such a product applied to any fixed vector is asymptotically gaussian the fluctuations we find can be thought of as a finite temperature correction to the limit in which first the size and then the number of matrices tend to infinity depending on the scaling limit considered the mean and variance of the limiting gaussian depend only on either the first two or the first four moments of the measure from which matrix entries are drawn we also obtain explicit error bounds on the moments of the norm and the kolmogorov smirnov distance to a gaussian finally we apply our result to obtain precise information about the stability of gradients in randomly initialized deep neural networks with relu activations this provides a quantitative measure of the extent to which the exploding and vanishing gradient problem occurs in a fully connected neural network with relu activations and a given architecture

noncommutative phase and unitarization of glp q 2 in this article imposing hermitian conjugate relations on the two parameter deformed quantum group glp q 2 is studied this results in a noncommutative phase associated with the unitarization of the quantum group after the achievement of the quantum group up q 2 with pq real via a noncommutative phase the representation of the algebra is built by means of the action of the operators constituting the up q 2 matrix on states

on probabilities of large deviations in banach spaces letx x 1 x 2 b denote a sequence of i i d random variables of a real separable banach spaceb y b denote a gaussian random variable suppose thatex ey 0 and that covariances ofx andy coincide denotes n n 1 2 x 1 x n we prove that under appropriate conditions p left s n r right p left y r right left 1 o left 1 right right as n to infty and give estimates of the remainder term applications to the 2 the anderson darling test and to the empirical characteristic functions are given

precise rates in the law of logarithm for i i d random variables let x x n n 1 be a sequence of i i d random variables set s n x 1 x 2 x n and m n max k n s k n 1 by using the strong approximation method we obtain that for any 1 email protected email protected n 2e n 2 b 2 email protected k 0 1 k 2k 1 2 b 2 if and only if ex 0 and ex 2 which strengthen and extend the result of gut and email protected taru 1 where n is the standard normal random variable furthermore l 2 convergence and a s convergence are also discussed

exact rates of convergence in some martingale central limit theorems abstract renz 14 ouchti 13 el machkouri and ouchti 3 and mourrat 12 have established some tight bounds on the rate of convergence in the central limit theorem for martingales in the present paper a modification of the methods developed by bolthausen 1 and grama and haeusler 7 is applied for obtaining exact rates of convergence in the central limit theorem for martingales with differences having conditional moments of order 2 0 our results generalise and strengthen the bounds mentioned above

duality for the matrix quantum group glp q 2 c it is shown that the algebra up q dual to glp q 2 c is isomorphic to u pq 1 2 sl 2 c z as a commutation algebra where z is a subalgebra central in up q the subalgebra z is a hopf subalgebra of up q while the commutation subalgebra u pq 1 2 sl 2 c is not a hopf subalgebra

malliavin calculus and asymptotic expansion for martingales we present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale the emphasis is put on the use of the malliavin calculus the uniform nondegeneracy of the malliavin covariance under certain truncation plays an essential role as the cramer condition did in the case of independent observations applications to statistics are presented

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

rates of convergence in the martingale central limit theorem we obtain a nonuniform estimate of the rate of convergence in the martingale central limit theorem for convergence to mixtures of normal distributions the uniform rates of convergence obtained by several other authors are special cases of our nonuniform estimate we also obtain a rate of convergence in b m brown s central limit theorem assuming only brown s elementary conditions this result is a martingale analogue of feller s generalization of the berry esseen theorem

kantorovich distance in the martingale clt and quantitative homogenization of parabolic equations with random coefficients the article begins with a quantitative version of the martingale central limit theorem in terms of the kantorovich distance this result is then used in the study of the homogenization of discrete parabolic equations with random i i d coefficients for smooth initial condition the rescaled solution of such an equation once averaged over the randomness is shown to converge polynomially fast to the solution of the homogenized equation with an explicit exponent depending only on the dimension polynomial rate of homogenization for the averaged heat kernel with an explicit exponent is then derived similar results for elliptic equations are also presented

an exact rate of convergence in the functional central limit theorem for special martingale difference arrays since the topology of weak convergence of probability distributions on the borel field of the space c c 0 1 is metrizable it is natural to describe the speed of convergence in weak functional limit theorems by means of an appropriate metric using the metric proposed by prokhorov it is shown that under suitable conditions the rate of convergence in the functional central limit theorem for c valued partial sum processes based on martingale difference arrays is the same as in the special case of row wise independent random variables where this rate is known to be an optimal one

limit theorems for weighted samples with applications to sequential monte carlo methods in the last decade sequential monte carlo methods smc emerged as a key tool in computational statistics see e g sequential monte carlo methods in practice 2001 springer new york monte carlo strategies in scientific computing 2001 springer new york complex stochastic systems 2001 109 173 these algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles which are generated recursively despite many theoretical advances see e g j roy statist soc ser b 63 2001 127 146 ann statist 33 2005 1983 2021 feynman kac formulae genealogical and interacting particle systems with applications 2004 springer ann statist 32 2004 2385 2411 the large sample theory of these approximations remains a question of central interest in this paper we establish a law of large numbers and a central limit theorem as the number of particles gets large we introduce the concepts of weighted sample consistency and asymptotic normality and derive conditions under which the transformations of the weighted sample used in the smc algorithm preserve these properties to illustrate our findings we analyze smc algorithms to approximate the filtering distribution in state space models we show how our techniques allow to relax restrictive technical conditions used in previously reported works and provide grounds to analyze more sophisticated sequential sampling strategies including branching resampling at randomly selected times and so on

broadband multilevel fast multipole algorithm based on an approximate diagonalization of the green s function we present a broadband multilevel fast multipole algorithm mlfma for fast and efficient solutions of three dimensional multiscale problems involving large objects with dense discretizations the proposed solver is based on the approximate diagonalization of the green s function using scaled spherical and plane waves leading to stable interaction computations for arbitrarily short distances in terms of wavelength despite contradictory requirements on the scaling factor that limit the accuracy of the diagonalization the resulting low frequency scheme is extremely stable and easy to plug into conventional mlfma implementations by simply extending built in tree structures converting them into broadband solvers without needing dense programming efforts accuracy and efficiency of the broadband mlfma are demonstrated on canonical problems as well as on multiscale problems that cannot be solved efficiently via standard implementations of mlfma

deviation inequalities for martingales with applications using changes of probability measure developed by grama and haeusler 2000 18 we extend the deviation inequalities of lanzinger and stadtmuller 2000 26 and fuk and nagaev 1971 15 to the case of martingales our inequalities recover the best possible decaying rate in the independent case in particular these inequalities improve the results of lesigne and voln 2001 27 under a stronger condition that the martingale differences have bounded conditional moments applications to linear regressions with martingale difference innovations weak invariance principles for martingales and self normalized deviations are provided in particular we establish a type of self normalized deviation bounds for parameter estimation of linear regressions such type bounds have the advantage that they do not depend on the distribution of the regression random variables

on the rate of convergence in the central limit theorem for martingales with discrete and continuous time heyde and brown 1970 established a bound on the rate of convergence in the central limit theorem for discrete time martingales having finite moments of order 2 28 with 0 0 moreover an example is constructed demonstrating that this bound is asymptotically exact for all 8 0 the result for discrete time martingales is then used to derive the corresponding bound on the rate of convergence in the central limit theorem for locally square integrable martingales with continuous time

a nondirective plane wave mlfma stable at low frequencies a novel method called the nondirective stable plane wave multilevel fast multipole algorithm nspwmlfma is presented to evaluate the low frequency lf interactions that cannot be handled by the multilevel fast multipole algorithm mlfma it is well known that the mlfma cannot be used for lf interactions since it suffers from numerical instability contrary to current techniques the proposed technique is not based on the spectral representation of the green function instead the addition theorem of the mlfma is manipulated into a form that allows numerically stable translations along the z axis the translation operator for these translations is derived in closed form a qr based method is devised to allow stable translations in all the other directions interpolations and anterpolations are also provided allowing a full multilevel algorithm since the nspwmlfma is based on the same mathematical foundations as the mlfma it requires limited adaptations to existing mlfma codes the fact that a qr is needed limits this algorithm to lf interactions however a coupling with the mlfma is straightforward allowing the easy construction of a broadband algorithm the dc limit of the algorithm is also presented and it is shown that the algorithm remains valid for static problems finally it is shown that the error introduced in the different steps of the algorithm is controllable and a single level vectorial version of the algorithm is applied to a generic scattering application to demonstrate its validity

precise asymptotics in complete moment convergence for self normalized sums let x x 1 x 2 be a sequence of nondegenerate i i d random variables with zero means set s n x 1 x n and v n 2 x 1 2 x n 2 ex 2i x x is a slowly varying function at we prove that for any b 2 d 2 b 1 email protected email protected b d 1 email protected n 2 logn d 2 bne s n v n 2i s n email protected n logn 1 b email protected n b d 1 b d 1 2 where n is a standard normal random variable

precise asymptotics in the baum katz and davis laws of large numbers let x x1 x2 be a sequence of i i d random variables such that ex 0 let z be a random variable possessing a stable distribution g with exponent 1 2 assume the distribution of x is attracted to g and set sn x1 xn we prove that n 1nr p 2p sn en1 p e p p r p 1 pr pe z p p r p 1 as e 0 for 1 p r under the additional assumption that there is normal attraction to g and that n 11np sn en1 p p p logeas e 0 for 1 p we close with a related result for the limiting case r p 2

efficient fast multipole method for low frequency scattering the solution of the helmholtz and maxwell equations using integral formulations requires to solve large complex linear systems a direct solution of those problems using a gauss elimination is practical only for very small systems with few unknowns the use of an iterative method such as gmres can reduce the computational expense most of the expense is then computing large complex matrix vector products the cost can be further reduced by using the fast multipole method which accelerates the matrix vector product for a linear system of size n the use of an iterative method combined with the fast multipole method reduces the total expense of the computation to n log n there exist two versions of the fast multipole method one which is based on a multipole expansion of the interaction kernel exp ikr r and which was first proposed by v rokhlin and another based on a plane wave expansion of the kernel first proposed by w c chew in this paper we propose a third approach the stable plane wave expansion spw fmm which has a lower computational expense than the multipole expansion and does not have the accuracy and stability problems of the plane wave expansion the computational complexity is n log n as with the other methods

a wideband fast multipole method for the helmholtz kernel this work presents a new fast multipole method fmm based on plane wave expansions pwfmm combining the advantages of the low and high frequency formulations we revisit the method of greengard etal 1998 devoted to the low frequency regime and based on the splitting of the greens function into a propagative and an evanescent part more precisely we give an explicit formula of the filtered translation function for the propagative part we derive a new formula for the evanescent part and we provide a new interpolation algorithm at all steps we check the accuracy of the method by providing error estimates these theoretical developments are used to propose a wideband fmm based entirely on plane wave expansions the numerical efficiency and accuracy of this broadband pwfmm are illustrated with a numerical example

computation and storage of the bianisotropic scalar green s function and its derivatives a set of algorithms is proposed for the accurate and efficient computation and storage of the bianisotropic scalar green s function the computation is based on an expansion of the green s function into chebyshev polynomials the analytical properties of these polynomials are exploited to allow the accurate computation of the derivatives of the green s function as well as the green s function itself for lossy materials the proposed computation strategy is provably robust in addition a multilevel storage scheme with a favorable complexity based on the chebyshev polynomial expansion is proposed for the storage of the expansion coefficients numerical results showcase the accuracy and computational complexity of the proposed algorithms

martingales and character ratios some general connections between martingales and character ratios of finite groups are developed as an application we sharpen the convergence rate in a central limit theorem for the character ratio of a random representation of the symmetric group on transpositions a generalization of these results is given for jack measure on partitions we also give a probabilistic proof of a result of burnside and brauer on the decomposition of tensor products

on moderate deviations for martingales let x n x t n f t n 0 t 1 be square integrable martingales with the quadratic characteristics x n n 1 2 we prove that the large deviations relation p x 1 n r 1 r 1 holds true for r growing to infinity with some rate depending on l 2 n z 0 t 1 x t n 2 28 and n 2 n e x n 1 1 1 where 0 and l 2 n 0 n 2 n 0 as n the exact bound for the remainder is also obtained

law of large numbers and central limit theorem for randomly forced pde s we consider a class of dissipative pde s perturbed by an external random force under the condition that the distribution of perturbation is sufficiently non degenerate a strong law of large numbers slln and a central limit theorem clt for solutions are established and the corresponding rates of convergence are estimated it is also shown that the estimates obtained are close to being optimal the proofs are based on the property of exponential mixing for the problem in question and some abstract slln and clt for mixing type markov processes

multiple decorrelation and rate of convergence in multidimensional limit theorems for the prokhorov metric the motivation of this work is the study of the error term et x in the averaging method for differential equations perturbed by a dynamical system results of convergence in distribution for frac e t varepsilon x cdot sqrt varepsilon varepsilon 0 have been established in khas minskii theory probab appl 11 1966 211 228 kifer ergodic theory dynamical systems 15 1995 1143 1172 and pene esaim probab statist 6 2002 33 88 we are interested here in the question of the rate of convergence in distribution of the family of random variables frac e t varepsilon x cdot sqrt varepsilon varepsilon 0 when goes to 0 t 0 and x in bf r d being fixed we will make an assumption of multiple decorrelation property satisfied in several situations we start by establishing a simpler result the rate of convergence in the central limit theorem for regular multidimensional functions in this context we prove a result of convergence in distribution with rate of convergence in o n 1 2 for all 0 for the prokhorov metric this result can be seen as an extension of the main result of pene comm math phys 225 2002 91 119 to the case of d dimensional functions in a second time we use the same method to establish a result of convergence in distribution for frac e t varepsilon x cdot sqrt varepsilon varepsilon 0 with rate of convergence in o 1 2 for the prokhorov metric we close this paper with a discussion in the appendix about the behavior of the quantity vert sup 0 le t le t 0 vert e t varepsilon x cdot vert infty vert l p under less stringent hypotheses

sur le theoreme de berry esseen pour les suites faiblement dependantes nous etendons la methode de demonstration du theoreme de berry esseen proposee par bergstrom aux suites de variables aleatoires faiblement dependantes en particulier nous montrons que pour les suites stationnaires de variables aleatoires reelles bornees la vitesse de convergence dans le theoreme limite central en distance de levy est de l ordre den 1 2 des que la suite p p 0 des coefficients de melange uniforme satisfait la condition p 0 p p

a low frequency efie mlfma solver based on approximate diagonalization of green s function the low frequency fast multipole algorithm lffma based on the approximate diagonalization of green s function for the electric field integral equation efie is proposed for open and complex structures with high accuracy moreover the multiresolution preconditioner is employed to the mixed potential form of efie with rwg basis functions for stability herein a triangle grouping scheme is adopted for the scalar potential so that the accuracy of the lffma can be guaranteed furthermore the computational resources can be reduced largely since part of the radiation pattern of vector potential can share one of the scalar potentials by taking advantage of some mathematical deductions

extension of the q deformed oscillator and two parameter coherent states the suq 2 algebra is extended by introducing additional raising and lowering operators and constructing their coherent states this new algebra of coherent states and the commutation relations between the extended operators are investigated and a resolution of unity is proposed

performing interpolation and anterpolation entirely by fast fourier transform in the 3 d multilevel fast multipole algorithm the fast multipole methods are used for solving a scalar acoustic or vector electromagnetic wave equation by integral equation methods with a large number of unknowns in this paper a new method is presented for performing interpolation and anterpolation in both spherical coordinates theta and phi by fft in the three dimensional 3 d multilevel fast multipole algorithm mlfma the key idea is to approximate functions on the unit sphere by truncated fourier series in two variables rather than by the usual spherical harmonics the proposed method is exact in interpolating and anterpolating and has the high numerical efficiency of fft the method is numerically compared to the method of performing interpolation and anterpolation using lagrangian interpolation which presently is probably the fastest method for those operations and the results suggest that the proposed new method is equally or more efficient depending on the desired accuracy

a mixed form fast multipole algorithm the fast multipole algorithm manifests in two very different forms at low frequencies and at mid frequencies each can operate in their respective regimes but are not tenable in the other regimes the paper reports on a way to factorize the green s function for fast algorithm using a mixed form the low frequency fast multipole algorithm lf fma will be used at low frequencies or the long wavelength regime and the multilevel fast multipole algorithm mlfma will be used for the mid frequencies or the shorter wavelength regime for object modeling where both long wavelength and wave physics are important we propose a mixed form fast multipole algorithm mf fma this algorithm has no low frequency break down and it can work seamlessly from static where circuit physics is important to dynamic where wave physics is important

finite dimensional representations of the quantum group glp q 2 using the exponential map from up q gl 2 using the fronsdal galindo formula for the exponential mapping from the quantum algebra up q gl 2 to the quantum group glp q 2 we show how the 2j 1 dimensional representations of glp q 2 can be obtained by exponentiating the well known 2j 1 dimensional representations of up q gl 2 for j 1 3 2 j 1 2 corresponds to the defining two dimensional t matrix the earlier results on the finite dimensional representations of glq 2 and slq 2 or suq 2 are obtained when p q representations of uq q 2 q in c mod r and uq 2 q in r mod 0 are also considered the structure of the clebsch gordan matrix for up q gl 2 is studied the same clebsch gordan coefficients are applicable in the reduction of the direct product representations of the quantum group glp q 2

precise rates in the law of logarithm for the moment convergence of i i d random variables abstract let x x n n 1 be a sequence of i i d random variables with e x 0 and e x 2 2 set s n k 1 n x k m n max k n s k n 1 let r 1 then we obtain lim e r 1 1 log e 2 r 1 n 1 n r 2 1 2 e m n e 2 n log n 2 r 1 2 holds if and only if e x 0 e x 2 2 and e x 2 r log x r

a low frequency stable plane wave addition theorem the multilevel fast multipole algorithm mlfma is a well known and very successful method for accelerating the matrix vector products required for the iterative solution of helmholtz problems the mlfma has an important drawback namely its inability to handle scattering problems with a lot of subwavelength detail due to the low frequency lf breakdown of the mlfma there is a need to extend the mlfma to lf since alternative methods are less efficient multipole methods or hard to implement spectral methods in this paper a new addition theorem will be developed that does not suffer from an lf breakdown instead it suffers from a high frequency hf breakdown the new method relies on a novel set of distributions the so called pseudospherical harmonics closely related to the spherical harmonics these allow the discretization points and translation operators to be calculated in closed form hence the method presented in this paper allows the easy implementation of a method that is stable at lf furthermore a combination of the traditional mlfma and the new method allows for the construction of a broadband mlfma