correlation functions in the itzykson zuber model then point function for the integral over unitary matrices with itzykson zuber measure is reduced to the integral over the gelfand tzetlin table the integrand for genericn is given by linear exponential times the rational function forn 2 and in some cases forn 2 later in fast is the polynomial and this allows to give an explicit and simple expression for all 2 point and a set ofn point functions for the most generaln point function a simple linear differential equation is constructed

generalized partially linear single index models abstract the typical generalized linear model for a regression of a response y on predictors x z has conditional mean function based on a linear combination of x z we generalize these models to have a nonparametric component replacing the linear combination t 0x t 0z by 0 t 0x t 0z where 0 is an unknown function we call these generalized partially linear single index models gplsim the models include the single index models which have 0 0 using local linear methods we propose estimates of the unknown parameters 0 0 and the unknown function 0 and obtain their asymptotic distributions examples illustrate the models and the proposed estimation methodology

approximate standard errors in semiparametric models we consider semiparametric models with p regressor terms and q smooth terms we obtain an explicit expression for the estimate of the regression coefficients given by the back fitting algorithm the calculation of the standard errors of these estimates based on this expression is a considerable computational exercise we present an alternative approximate method of calculation that is less demanding with smoothing splines the method is exact while with loess it gives good estimates of standard errors we assess the adequacy of our approximation and of another approximation with the help of two examples

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

random walk models for multifractals reversible random walk models for bernoulli multifractals in dimension 2 together with marginal versions of these are first studied to account for both geometrical and analytical properties of such objects using large deviation theory it is also shown how the transience recurrence of a more complex reversible random walk is related to the minimum energy flow convergence divergence property new results on the asymptotics of such a walker are then derived in both transient recurrent situations

an algorithm for the minimization of mixed l sub 1 and l sub 2 norms with application to bayesian estimation the regularizing functional approach is widely used in many estimation problems in practice the solution is defined as one minimum point of a suitable functional the main part of which accounts for the underlying physical model whereas the regularizing part represents some prior information about the unknowns in the bayesian interpretation one has a maximum a posteriori map estimator in which the main and regularizing parts are represented respectively by likelihood and prior distributions when either the prior or likelihood is a laplace distribution and the other is a gaussian distribution one is led to consider functionals that include both absolute and square norms the authors present a characterization of the minimum points of such functionals together with a descent type algorithm for numerical computations the results of monte carlo simulations are also reported

random walks in random dirichlet environment are transient in dimension d 3 we consider random walks in random dirichlet environment rwde which is a special type of random walks in random environment where the exit probabilities at each site are i i d dirichlet random variables on z d rwde are parameterized by a 2d uplet of positive reals we prove that for all values of the parameters rwde are transient in dimension d ge 3 we also prove that the green function has some finite moments and we characterize the finite moments our result is more general and applies for example to finitely generated symmetric transient cayley graphs in terms of reinforced random walks it implies that directed edge reinforced random walks are transient for d ge 3

function theory random paths and covering spaces pour des varietes de riemann connexes m on etudie la relation entre la theorie de fonction harmonique sur m les proprietes statistiques des chemins aleatoires sur m et la structure geometrique globale de m

semiparametric inference in a partial linear model in a partial linear model the dependence of a response variate y on covariates w x is given by y w x e where e is independent of w x with densities g and f respectively in this paper an asymptotically efficient estimator of is constructed solely under mild smoothness assumptions on the unknown f and g thereby removing the assumption of finite residual variance on which all least squares type estimators available in the literature are based

isoperimetric inequalities and transient random walks on graphs recurrent we show that any graph satisfying an isoperimetric inequality only slightly stronger than that of z2 is transient more precisely if f k denotes the smallest number of vertices in the boundary of a connected subgraph with k vertices then the graph is transient if the infinite sum e f k 2 converges this can be applied to parabolicity versus hyperbolicity of surfaces 1 introduction let g be a connected graph which is locally finite that is all vertices have finite degree we consider a random walk starting at a vertex v say such that at any vertex u the walk proceeds to a neighbour with probability 1 d u where d u is the degree i e the number of neighbours of u the graph g is recurrent if we revisit v with probability 1 otherwise g is transient it is well known that the three dimensional grid z3 is transient while z2 is recurrent more generally nash williams 13 see also 4 10 proved that any graph with smaller growth rate than z2 is recurrent lyons 10 showed that certain subgraphs of grids are transient provided they grow just a little faster than z2 other results in terms of isoperimetric inequalities supporting the statement that z2 is in a sense an extreme recurrent graph can be derived from work of fernandez 5 grigor yan 7 and varopoulos 15 varopoulos 16 used results of gromov 8 to characterize completely the recurrent cayley graphs we shall carry these results further if v is a vertex set in g then av will denote the boundary of v that is the set of vertices of v having neighbours outside of v let f be a nondecreasing positive real function defined on the natural numbers we say that g satisfies an f isoperimetric inequality if there exists a constant c 0 such that for each finite vertex set v of g

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

differential systems for biorthogonal polynomials appearing in 2 matrix models and the associated riemann hilbert problem we consider biorthogonal polynomials that arise in the study of a generalization of two matrix hermitian models with two polynomial potentials v1 x v2 y of any degree with arbitrary complex coefficients finite consecutive subsequences of biorthogonal polynomials windows of lengths equal to the degrees of the potentials v1 and v2 satisfy systems of ode s with polynomial coefficients as well as pde s deformation equations with respect to the coefficients of the potentials and recursion relations connecting consecutive windows a compatible sequence of fundamental systems of solutions is constructed for these equations the stokes sectorial asymptotics of these fundamental systems are derived through saddle point integration and the riemann hilbert problem characterizing the differential equations is deduced

random walk on the infinite cluster of the percolation model we consider random walk on the infinite cluster of bond percolation on d we show that in the supercritical regime whend 3 this random walk is a s transient this conclusion is achieved by considering the infinite percolation cluster as a random electrical network in which each open edge has unit resistance it is proved that the effective resistance of this network between a nominated point and the points at infinity is almost surely finite

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

wavelet generalized least squares a new blu estimator of linear regression models with 1 f errors long memory noise is common to many areas of signal processing and can seriously confound estimation of linear regression model parameters and their standard errors classical autoregressive moving average arma methods can adequately address the problem of linear time invariant short memory errors but may be inefficient and or insufficient to secure type 1 error control in the context of fractal or scale invariant noise with a more slowly decaying autocorrelation function here we introduce a novel method called wavelet generalized least squares wls which is to a good approximation the best linear unbiased blu estimator of regression model parameters in the context of long memory errors the method also provides maximum likelihood ml estimates of the hurst exponent which can be readily translated to the fractal dimension or spectral exponent characterizing the correlational structure of the errors and the error variance the algorithm exploits the whitening or kar

local strong homogeneity of a regularized estimator this paper deals with regularized pointwise estimation of discrete signals which contain large strongly homogeneous zones where typically they are constant or linear or more generally satisfy a linear equation the estimate is defined as the minimizer of an objective function combining a quadratic data fidelity term and a regularization prior term the latter term is the sum of the values obtained by applying a potential function pf to each component called a difference of a linear transform of the signal minimizers of functions of this form arise in various settings in statistics and optimization the features exhibited by such an estimate are closely related to the shape of the pf our goal is to determine estimators providing solutions which involve large strongly homogeneous zones where more precisely the differences are null in spite of the noise corrupting the data to this end we require that the strongly homogeneous zones recovered by the estimator be insensitive to any variation of th

activation detection in functional mri using subspace modeling and maximum likelihood estimation a statistical method for detecting activated pixels in functional mri fmri data is presented in this method the fmri time series measured at each pixel is modeled as the sum of a response signal which arises due to the experimentally controlled activation baseline pattern a nuisance component representing effects of no interest and gaussian white noise for periodic activation baseline patterns the response signal is modeled by a truncated fourier series with a known fundamental frequency but unknown fourier coefficients the nuisance subspace is assumed to be unknown a maximum likelihood estimate is derived for the component of the nuisance subspace which is orthogonal to the response signal subspace an estimate for the order of the nuisance subspace is obtained from an information theoretic criterion a statistical test is derived and shown to be the uniformly most powerful ump test invariant to a group of transformations which are natural to the hypothesis testing problem the maximal invariant statistic used in this test has an f distribution the theoretical f distribution under the null hypothesis strongly concurred with the experimental frequency distribution obtained by performing null experiments in which the subjects did not perform any activation task applications of the theory to motor activation and visual stimulation fmri studies are presented

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

penalized partially linear models using sparse representations with an application to fmri time series in this paper we consider modeling the nonparametric component in partially linear models plms using linear sparse representations e g wavelet expansions two types of representations are investigated namely orthogonal bases complete and redundant overcomplete expansions for bases we introduce a regularized estimator of the nonparametric part the important contribution here is that the nonparametric part can be parsimoniously estimated by choosing an appropriate penalty function for which the hard and soft thresholding estimators are special cases this allows us to represent in an effective manner a broad class of signals including stationary and or nonstationary signals and avoids excessive bias in estimating the parametric component we also give a fast estimation algorithm the method is then generalized to handle the case of overcomplete representations a large scale simulation study is conducted to illustrate the finite sample properties of the estimator the estimator is finally applied to real neurophysiological functional magnetic resonance imaging mri data sets that are suspected to contain both smooth and transient drift features

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

eaton s markov chain its conjugate partner and mathscr p admissibility suppose that x is a random variable with density f x and that x is a proper posterior corresponding to an improper prior the prior is called p admissible if the generalized bayes estimator of every bounded function of is almost admissible under squared error loss eaton showed that recurrence of the markov chain with transition density r x f x dx is a sufficient condition for p admissibility of we show that eaton s markov chain is recurrent if and only if its conjugate partner with transition density r y x f y x d is recurrent this provides a new method of establishing p admissibility often one of these two markov chains corresponds to a standard stochastic process for which there are known results on recurrence and transience for example when x is poisson and an improper gamma prior is placed on the markov chain defined by r y x is equivalent to a branching process with immigration we use this type of argument to establish p admissibility of some priors when f is a negative binomial mass function and when f is a gamma density with known shape

spline smoothing in a partly linear model on donne des estimations qui minimisent la somme des sommes residuelles des carres et une penalite de difficulte

two matrix model with semiclassical potentials and extended whitham hierarchy we consider the two matrix model with potentials whose derivatives are arbitrary rational functions of fixed pole structure and the support of the spectra of the matrices are union of intervals hard edges we derive an explicit formula for the planar limit of the free energy and we derive a calculus which allows us to compute derivatives of arbitrarily high order by extending classical rauch s variational formulae the four point correlation functions are explicitly worked out the formalism extends naturally to the computation of residue formulae for the tau function of the so called universal whitham hierarchy studied mainly by i krichever our setting extends the moduli space in that there are certain extra data

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

testing the goodness of fit of a linear model via nonparametric regression techniques abstract this article investigates the use of nonparametric regression methodology to test the adequacy of a parametric linear model the large sample properties of parametric goodness of fit tests for linearity are considered the inadequacies of such tests lead to the proposal of new tests that are constructed from nonparametric regression fits to the residuals from linear regression large sample theory is derived for two variants of this type of statistic the results demonstrate that such tests are consistent against all fixed smooth alternatives to linearity but are incapable of detecting local alternatives converging to a linear model at the parametric rate n 1 2 simulation experiments involving a test based on fitting cubic smoothing splines to residuals reveals that this test has good power properties against several reasonable alternatives

multifractal formalism for self similar bridges we derive the thermodynamics of self similar paths or bridges joining the two points and of the plane these paths may be constituted with both macroscopic and microscopic fragments each deserving its specific statistics while remaining continuous such discontinuous paths are also studied with some information related to the statistics of their jumps if the bridges under study are bound to be non decreasing paths this study coincides with the one of multifractal measures on the unit interval relaxing this condition leads to an extension of the multifractal formalism whose main lines are derived here

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

the dimension spectrum of some dynamical systems we analyze the dimension spectrum previously introduced and measured experimentally by jensen kadanoff and libchaber using large deviation theory we prove for some invariant measures of expanding markov maps that the hausdorff dimensionf of the set on which the measure has a singularity is a well defined concave and regular function in particular we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number we also show in these particular cases that the functionf is universal

critical phenomena for spitzer s reversible nearest particle systems motivated by several results and open problems concerning harris basic contact process we consider the relationship between the critical behavior of the finite and infinite versions of spitzer s reversible nearest particle systems we show that the critical values for the finite and infinite systems agree but that the behavior of the two systems at the common critical value can differ the nash williams recurrence criterion for reversible markov chains is an important tool used in the proofs of the main results and we give a new treatment of that theory finally we compute several critical exponents for the nearest particle systems

convergence rates for parametric components in a partly linear model consider the regression model y i x i beta g t i e i for i 1 cdots n here g is an unknown holder continuous function of known order p in r beta is a k times 1 parameter vector to be estimated and e i is an unobserved disturbance such a model is often encountered in situations in which there is little real knowledge about the nature of g a piecewise polynomial g n is proposed to approximate g the least squares estimator hat beta is obtained based on the model y i x i beta g n t i e i it is shown that hat beta can achieve the usual parametric rates n 1 2 with the smallest possible asymptotic variance for the case that x and t are correlated

quasi likelihood estimation in semiparametric models abstract suppose the expected value of a response variable y may be written h x t where x and t are covariates each of which may be vector valued is an unknown parameter vector is an unknown smooth function and h is a known function in this article we outline a method for estimating the parameter of this type of semiparametric model using a quasi likelihood function algorithms for computing the estimates are given and the asymptotic distribution theory for the estimators is developed the generalization of this approach to the case in which y is a multivariate response is also considered the methodology is illustrated on two data sets and the results of a small monte carlo study are presented

2 matrix versus complex matrix model integrals over the unitary group as triangular integrals we prove that the 2 hermitian matrix model and the complex matrix model obey the same loop equations and as a byproduct we find a formula for itzykzon zuber type integrals over the unitary group integrals over u n are rewritten as gaussian integrals over triangular matrices and then computed explicitly that formula is an efficient alternative to the former shatashvili s formula

mixed correlation function and spectral curve for the 2 matrix model we compute the mixed correlation function in a way which involves only the orthogonal polynomials with degrees close to n in some sense like the christoffel darboux theorem for non mixed correlation functions we also derive new representations for the differential systems satisfied by the bi orthogonal polynomials and we find new formulae for the spectral curve in particular we prove the conjecture of m bertola claiming that the spectral curve is the same curve which appears in the loop equations

kernel smoothing in partial linear models on considere deux methodes d estimation l une reliee aux splines de lissage partiels l autre motivee par une analyse de residus partielle

unpredictable nearest neighbor processes benjamini pemantle and peres constructed nearest neighbor processes which have predictability profiles that decay faster than that of the simple random walk haggstrom and mossel found processes with even faster decaying predictability profiles we prove that the rate of decay achieved by haggstrom and mossel is optimal

a method of integration over matrix variables the integral over twon n hermitan matricesz g c dadbexp tr a 2 b 2 2cab g n a 4 b 4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

wavelet based estimation of a semiparametric generalized linear model of fmri time series addresses the problem of detecting significant changes in fmri time series that are correlated to a stimulus time course this paper provides a new approach to estimate the parameters of a semiparametric generalized linear model of the fmri time series the fmri signal is described as the sum of two effects a smooth trend and the response to the stimulus the trend belongs to a subspace spanned by large scale wavelets the wavelet transform provides an approximation to the karhunen loeve transform for the long memory noise and we have developed a scale space regression that permits one to carry out the regression in the wavelet domain while omitting the scales that are contaminated by the trend in order to demonstrate that our approach outperforms the state of the art detrending technique we evaluated our method against a smoothing spline approach experiments with simulated data and experimental fmri data demonstrate that our approach can infer and remove drifts that cannot be adequately represented with splines

mixed correlation functions of the two matrix model we compute the correlation functions mixing the powers of two non commuting random matrices within the same trace the angular part of the integration was partially known in the literature morozov a 1992 mod phys lett a 7 3503 shatashvili s l 1993 commun math phys 154 421 we pursue the calculation and carry out the eigenvalue integration reducing the problem to the construction of the associated bi orthogonal polynomials the generating function of these correlations then becomes a determinant involving the recursion coefficients of the bi orthogonal polynomials

multifractal like features of the paths probability distribution in the two dimensional directed random walk abstract we consider the two dimensional directed random walk with probabilities px and py assigned to steps in x and y direction respectively and analyse the probability distribution for the possible 2t paths the walker may follow when it performs a t step walk this distribution is multifractal since its qth moment has the typical power law behaviour of multifractal distributions unless p x p y 1 2 if the values of px and py are allowed to fluctuate around their average values and throughout the lattice the qth moment averaged over the possible realizations of the lattice exhibits the multifractal power law behaviour even when x y 1 2 the existence of negative fractal dimensions is also analysed

curve fitting by polynomial trigonometric regression summary we consider a method of estimating an unknown regression curve by regression on a combination of low order polynomial terms and trigonometric terms estimation based on trigonometric functions alone is known to suffer from bias problems at the boundaries due to the periodic nature of the fitted functions we show that these boundary problems are alleviated by adding low order polynomial terms the utility of the method is illustrated with examples and asymptotic results show that the estimators are competitive with other nonparametric procedures applications to estimation in semiparametric models are also considered with particular emphasis on the case of analysis of covariance when the relationship between the response and the covariate is difficult to model parametrically