correlations between zeros of a random polynomial we obtain exact analytical expressions for correlations between real zeros of the kac random polynomial we show that the zeros in the interval 1 1 are asymptotically independent of the zeros outside of this interval and that the straightened zeros have the same limit translation invariant correlations then we calculate the correlations between the straightened zeros of theo 1 random polynomial

universal fluctuations of zeros of chaotic wavefunctions wavefunctions of one and two dimensional quantum systems can be parametrized by a finite number of zeros lying in phase space we study correlations of these zeros for fully chaotic systems in terms of a statistical model based on random polynomials excellent agreement is found for the two point correlation function and nearest neighbour spacing distribution of this model and the results obtained for wavefunctions of dynamical systems we conjecture that these correlation functions are valid for any chaotic system after rescaling the phase space distances unfolding some consequences for the distribution of zeros due to time reversal symmetry are also discussed

random rates in anisotropic regression in the context of minimax theory we propose a new kind of risk normalized by a random variable measurable with respect to the data we present a notion of optimality and a method to construct optimal procedures accordingly we apply this general setup to the problem of selecting significant variables in gaussian white noise in particular we show that our method essentially improves the accuracy of estimation in the sense of giving explicit improved confidence sets in l2 norm links to adaptive estimation are discussed 1 introduction searching for significant variables is certainly one of the oldest and most popular problems in statistics one of the simplest models where the issue of selecting significant variables was first stated mathematically is linear regression a vast literature has been devoted to this topic since and different approaches have been proposed over the last forty years both for estimation and for hypothesis testing among many authors we refer to akaike 1 breiman and freedman 3 chernoff 5 csiszar and korner 6 dychakov 10 patel 42 renyi 46 freidlina 13 meshalkin 35 malyutov and tsitovich 34 schwarz 47 and stone 48 in classical parametric regression if we consider a linear model we first have to measure the possible gain of searching for a limited number of significant variables if the model comes from a specific field of application then only an adequate description together with its solution is relevant however from a mathematical point of view a theory of selecting significant variables does not lead at least asymptotically to a substantial improvement of the accuracy of estimation in a regular parametric model the classical n rate of convergence is not affected by the number of significant variables however even in this setup let us emphasize that asymptotically has to be understood as up to a constant and that the correct choice of significant variables may possibly improve this constant if instead of a linear model we consider a nonparametric regression model the search for significant variables becomes crucial for estimating the regression function the rate of convergence explicitly depends on the set of significant variables let us develop this statement with the following example of multivariate regression suppose we observe z n xi y i i 1 n in the model

density estimation by wavelet thresholding density estimation is a commonly used test case for nonparametric estimation methods we explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients minimax rates of convergence are studied over a large range of besov function classes b pq and for a range of global l p error measures 1 p p some form of nonlinearity is essential since the minimax linear estimators are suboptimal by polynomial powers of n a second approach using an approximation of a gaussian white noise model in a mallows metric is used to attain exactly optimal rates of convergence for quadratic error p 2

on coverage and local radial rates of credible sets in the mildly ill posed inverse signal in white noise model we construct confidence sets as credible balls with respect to the empirical bayes posterior resulting from a certain two level hierarchical prior the quality of the posterior is characterized by the contraction rate which we allow to be local that is depending on the parameter the issue of optimality of the constructed confidence sets is addressed via a trade off between its size the local radial rate and its coverage probability we introduce excessive bias restriction ebr more general than self similarity and polished tail condition recently studied in the literature under ebr we establish the confidence optimality of our credible set with some local oracle radial rate we also derive the oracle estimation inequality and the oracle posterior contraction rate the obtained local results are more powerful than global adaptive minimax results for a number of smoothness scales follow as consequence in particular the ones considered by szabo et al ann statist 43 2015 1391 1428

the complex zeros of random polynomials mark kac gave an explicit formula for the expectation of the number vn a of zeros of a random polynomial n i pn z e tj j o in any measurablc subset q of the reals here in i are independent standard normal random variables in fact for each n 1 he obtained an explicit intensity function gn for which e vn l jgn x dx here we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset q of the complex plane c namely we show that e vn ki j hn x y dxdy j gn x dx where hn is an explicit intensity function we also study the asymptotics of hn showing that for large n its mass lies close to and is uniformly distributed around the unit circle

global uniform risk bounds for wavelet deconvolution estimators we consider the statistical deconvolution problem where one observes n replications from the model y x where x is the unobserved random signal of interest and is an independent random error with distribution under weak assumptions on the decay of the fourier transform of we derive upper bounds for the finite sample sup norm risk of wavelet deconvolution density estimators f n for the density f of x where f r r is assumed to be bounded we then derive lower bounds for the minimax sup norm risk over besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds we further show that linear estimators adapt to the unknown smoothness of f if the fourier transform of decays exponentially and that a corresponding result holds true for the hard thresholding wavelet estimator if decays polynomially we also analyze the case where f is a supersmooth analytic density we finally show how our results and recent techniques from rademacher processes can be applied to construct global confidence bands for the density f

chaotic analytic zero points exact statistics for those of a random spin state a natural statistical ensemble of 2j points on the unit sphere can be associated via the majorana representation with a random quantum state of spin j and an exact expression is obtained here for the general k point correlation function in this ensemble the pair correlation in the large j limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere it appears from the numerical work of others that in this limit these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems

distribution of roots of random polynomials we consider polynomials of high degree with random coefficients which appear in the context of quantum chaotic dynamics and investigate various conditions under which their roots tend to concentrate near the unit circle in the complex plane correlation functions of roots are computed analytically we also investigate a certain class of random polynomials whose roots cover in a uniform way the riemann sphere special emphasis is devoted to the influence of symmmetries

phase space approach to quantum dynamics replaces the schrodinger equation for the time propagation of states of a quantized 2d spherical phase space by the dynamics of a system of n particles lying in phase space this is done through factorization formulae of an analytic function theory arising in a coherent state representation the particles being the zeroes of the quantum state for linear hamiltonians like a spin in a uniform magnetic field the motion of the particles is classical however nonlinear terms induce interactions between the particles their time propagation is studied and it is shown that contrary to integrable systems for chaotic maps they tend to fill as their classical counterpart the whole phase space in a uniform way

rates of contraction of posterior distributions based on gaussian process priors we derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on gaussian processes the rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel hilbert space of the gaussian process and the small ball probabilities of the gaussian process we determine these quantities for a range of examples of gaussian priors and in several statistical settings for instance we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a fractionally integrated brownian motion we also consider regression with gaussian errors and smooth classification under a logistic or probit link function combined with various priors

exact statistics of complex zeros for gaussian random polynomials with real coefficients k point correlations of complex zeros for gaussian ensembles of random polynomials of order n with real coefficients grprc are calculated exactly following an approach of hannay for the case of gaussian random polynomials with complex coefficients grpcc it is shown that in the thermodynamic limit of gaussian random holomorphic functions all the statistics converge to their grpcc counterparts as one moves off the real axis while close to the real axis the two cases are essentially different special emphasis is given to one and two point correlation functions in various regimes

the smoothness test for a density function abstract the problem of testing hypothesis that a density function has no more than derivatives versus it has more than derivatives is considered for a solution the l 2 norms of wavelet orthogonal projections on some orthogonal differences of spaces from a multiresolution analysis is used for the construction of the smoothness test an asymptotic distribution of a smoothness estimator is used to analyze that asymptotic distribution a new technique of enrichment procedure is proposed the finite sample behavior of the smoothness test is demonstrated in a numerical experiment in case of determination if a density function is continuous or discontinuous

quantum chaotic dynamics and random polynomials we investigate the distribution of roots of polynomials of high degree with random coefficients which among others appear naturally in the context of quantum chaotic dynamics it is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane in order to further increase this tendency we study in detail the particular case of self inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle correlation functions of these roots are also computed analytically and compared to the correlations of eigenvalues of random matrices the problem of ergodicity of chaotic wavefunctions is also considered for that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space while these results are consistent with random matrix theory predictions they provide a new and different insight into the problem of quantum ergodicity special attention is devoted to the role of symmetries in the distribution of roots of random polynomials

statistical inference for the optimal approximating model in the setting of high dimensional linear models with gaussian noise we investigate the possibility of confidence statements connected to model selection although there exist numerous procedures for adaptive point estimation the construction of adaptive confidence regions is severely limited cf li in ann stat 17 1001 1008 1989 the present paper sheds new light on this gap we develop exact and adaptive confidence regions for the best approximating model in terms of risk one of our constructions is based on a multiscale procedure and a particular coupling argument utilizing exponential inequalities for noncentral 2 distributions we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one

a simple bootstrap method for constructing nonparametric confidence bands for functions standard approaches to constructing nonparametric confidence bands for functions are frustrated by the impact of bias which generally is not estimated consistently when using the bootstrap and conventionally smoothed function estimators to overcome this problem it is common practice to either undersmooth so as to reduce the impact of bias or oversmooth and thereby introduce an explicit or implicit bias estimator however these approaches and others based on nonstandard smoothing methods complicate the process of inference for example by requiring the choice of new unconventional smoothing parameters and in the case of undersmoothing producing relatively wide bands in this paper we suggest a new approach which exploits to our advantage one of the difficulties that in the past has prevented an attractive solution to the problem the fact that the standard bootstrap bias estimator suffers from relatively high frequency stochastic error the high frequency together with a technique based on quantiles can be exploited to dampen down the stochastic error term leading to relatively narrow simple to construct confidence bands

adaptive confidence bands for nonparametric regression functions this article proposes a new formulation for the construction of adaptive confidence bands cbs in nonparametric function estimation problems cbs which have size that adapts to the smoothness of the function while guaranteeing that both the relative excess mass of the function lying outside the band and the measure of the set of points where the function lies outside the band are small it is shown that the bands adapt over a maximum range of lipschitz classes the adaptive cb can be easily implemented in standard statistical software with wavelet support we investigate the numerical performance of the procedure using both simulated and real datasets the numerical results agree well with the theoretical analysis the procedure can be easily modified and used for other nonparametric function estimation models supplementary materials for this article are available online

definitions of quasiconformality summary we establish that the infinitesimal h definition for quasiconforreal mappings on carnot groups implies global quasisymmetry and hence the absolute continuity on almost all lines our method is new even in r where we obtain that the limsup condition in the h definition can be replaced by a liminf condition this leads to a new removability result for quasi conformal mappings in euclidean spaces an application to parametrizations of chord arc surfaces is also given

an exponential inequality for the distribution function of the kernel density estimator with applications to adaptive estimation it is shown that the uniform distance between the distribution function f n k h of the usual kernel density estimator based on an i i d sample from an absolutely continuous law on mathbb r with bandwidth h and the empirical distribution function fn satisfies an exponential inequality this inequality is used to obtain sharp almost sure rates of convergence of f n k h n f n infty under mild conditions on the range of bandwidths hn including the usual mise optimal choices another application is a dvoretzky kiefer wolfowitz type inequality for f n k h f infty where f is the true distribution function the exponential bound is also applied to show that an adaptive estimator can be constructed that efficiently estimates the true distribution function f in sup norm loss and at the same time estimates the density of f if it exists but without assuming it does at the best possible rate of convergence over holder balls again in sup norm loss

adaptive confidence sets in l 2 the problem of constructing confidence sets that are adaptive in l 2 loss over a continuous scale of sobolev classes of probability densities is considered adaptation holds where possible with respect to both the radius of the sobolev ball and its smoothness degree and over maximal parameter spaces for which adaptation is possible two key regimes of parameter constellations are identified one where full adaptation is possible and one where adaptation requires critical regions be removed techniques used to derive these results include a general nonparametric minimax test for infinite dimensional null and alternative hypotheses and new lower bounds for l 2 adaptive confidence sets

frequentist coverage of adaptive nonparametric bayesian credible sets we investigate the frequentist coverage of bayesian credible sets in a nonparametric setting we consider a scale of priors of varying regularity and choose the regularity by an empirical bayes method next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity we show that such an adaptive bayes credible set gives correct uncertainty quantification of polished tail parameters in the sense of high probability of coverage of such parameters on the negative side we show by theory and example that adaptation of the prior necessarily leads to gross and haphazard uncertainty quantification for some true parameters that are still within the hyperrectangle regularity scale

on adaptive posterior concentration rates tion of holder balls and that moreover achieve our lower bound we analyse the consequences in terms of asymptotic behaviour of poste rior credible balls as well as frequentist minimax adaptive estimation our results are appended with an upper bound for the contraction rate under an arbitrary loss in a generic regular experiment the up per bound is attained for certain sieve priors and enables to extend our results to density estimation

multiscale change point inference summary we introduce a new estimator the simultaneous multiscale change point estimator smuce for the change point problem in exponential family regression an unknown step function is estimated by minimizing the number of change points over the acceptance region of a multiscale test at a level the probability of overestimating the true number of change points k is controlled by the asymptotic null distribution of the multiscale test statistic further we derive exponential bounds for the probability of underestimating k by balancing these quantities will be chosen such that the probability of correctly estimating k is maximized all results are even non asymptotic for the normal case on the basis of these bounds we construct asymptotically honest confidence sets for the unknown step function and its change points at the same time we obtain exponential bounds for estimating the change point locations which for example yield the minimax rate up to a log term finally the simultaneous multiscale change point estimator achieves the optimal detection rate of vanishing signals as n even for an unbounded number of change points we illustrate how dynamic programming techniques can be employed for efficient computation of estimators and confidence regions the performance of the multiscale approach proposed is illustrated by simulations and in two cutting edge applications from genetic engineering and photoemission spectroscopy

anti concentration and honest adaptive confidence bands modern construction of uniform condence e and nickl 2010 this condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process equivalently for the supremum of a gaussian process with the same covariance function as that of the studentized empirical process the principal contribution of this paper is to remove the need for this classical condition we show that a considerably weaker sucient condi tion is derived from an anti concentration property of the supremum of the approximating gaussian process and we derive an inequality lead ing to such a property for separable gaussian processes we refer to the new condition as a generalized sbr condition our new result shows that the supremum does not concentrate too fast around any value we then apply this result to derive a gaussian multiplier bootstrap procedure for constructing honest condence bands for nonparametric density estimators this result can be applied in other nonparametric problems as well an essential advantage of our approach is that it ap plies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist or is un known this is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data driven fash ion which is needed for adaptive constructions of the condence bands furthermore our approach is asymptotically honest at a polynomial rate namely the error in coverage level converges to zero at a fast polynomial speed with respect to the sample size in sharp contrast the approach based on extreme value theory is asymptotically honest only at a logarithmic rate the error converges to zero at a slow loga rithmic speed finally of independent interest is our introduction of a new practical version of lepski s method which computes the optimal non conservative resolution levels via a gaussian multiplier bootstrap method

quasiconformal 4 manifolds for any pseudo group of homeomorphism of euclidean space one can define the corresponding category of manifolds the most familiar examples in topology are the full pseudo group of homeomorphisms giving rise to the theory of topological manifolds and the subgroup of smooth differomorphisms giving rise to the theory of c manifolds in this paper we discuss an intermediate category quasiconformal homeomorphisms and manifolds recall that a homeomorphism 9 d r n from a domain d in r n to its image 9 d is k quasiconformal if for all x in d

can one estimate the conditional distribution of post model selection estimators we consider the problem of estimating the conditional distribution of a post model selection estimator where the conditioning is on the selected model the notion of a post model selection estimator here refers to the combined procedure resulting from first selecting a model e g by a model selection criterion such as aic or by a hypothesis testing procedure and then estimating the parameters in the selected model e g by least squares or maximum likelihood all based on the same data set we show that it is impossible to estimate this distribution with reasonable accuracy even asymptotically in particular we show that no estimator for this distribution can be uniformly consistent not even locally this follows as a corollary to local minimax lower bounds on the performance of estimators for this distribution similar impossibility results are also obtained for the conditional distribution of linear functions e g predictors of the post model selection estimator

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

uniform limit theorems for wavelet density estimators let pn y k y k jn 1 l 0 k lk 2l 2 2 l y k be the linear wavelet density estimator where are a father and a mother wavelet with compact support are the empirical wavelet coefficients based on an i i d sample of random variables distributed according to a density p 0 on r and j n z j n several uniform limit theorems are proved first the almost sure rate of convergence of sup y r pn y ep n y is obtained and a law of the logarithm for a suitably scaled version of this quantity is established this implies that sup y r p n y po y attains the optimal almost sure rate of convergence for estimating p 0 if j n is suitably chosen second a uniform central limit theorem as well as strong invariance principles for the distribution function of p n that is for the stochastic processes n f w n s f s n s pn p 0 s r are proved and more generally uniform central limit theorems for the processes n p n p 0 f f for other donsker classes f of interest are considered as a statistical application it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by donoho et al ann statist 24 1996 508 539

exact statistical properties of the zeros of complex random polynomials the zeros of complex gaussian random polynomials with coefficients such that the density in the underlying complex space is uniform are known to have the same statistical properties as the zeros of the coherent state representation of one dimensional chaotic quantum systems we extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest landau level a study of the statistical properties of the zeros is undertaken using exact formulae for the one and two point distribution functions attention is focused on the moments of the two point correlation in the bulk the variance of a linear statistic and the asymptotic form of the two point correlation at the boundary a comparison is made with the same quantities for the eigenvalues of complex gaussian random matrices

factoring the logarithmic spiral distortion of length and angle are widely studied in geometry and dynamics much of the success in the study of one dimensional complex dynamics flow from the ahlfors bers sullivan a b s 1 theory for deforming conformal structures essentially from the existence and uniqueness theory for the beltrami equation d f d f no comparable analytic theory exists for the distortion of length in the plane or of length or angle in higher dimensions in these cases the local distortion is specified by more than d parameters in dimension d so the analogous equations are over determined what exists instead is a coarser more topological discussion initiated by sullivan of structures extension and isotopy of imbeddings s 2 t v in the quasiconformal and bilipschitz quasi isometric categories the coarseness is the use of the alexander isotopy which crushes rather than unwinds homeomorphisms in this paper we find a constraint on the more delicate method of returning a quasi isometry to the identity factorization into maps of small isometric distortion we study the following l quasi isometry of the closed unit disk a xl x2 x 2 x2 1

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions