some limit theorems for the empirical process indexed by functions let n 1 be a sequence of classes of real valued measurable functions defined on a probability space s p under weak metric entropy conditions on n 1 and under growth conditions on we show that there are non zero numerical constantsc1 andc2 such that where n is a non decreasing function ofn related to the metric entropy of a few applications of this general result are considered we obtain a s rates of uniform convergence for the empirical process indexed by intervals as well as a s rates of uniform convergence for the empirical characteristic function over expanding intervals

hyper sparse optimal aggregation given a finite set f of functions and a learning sample the aim of an aggregation procedure is to have a risk as close as possible to risk of the best function in f up to now optimal aggregation procedures are convex combinations of every elements of f in this paper we prove that optimal aggregation procedures combining only two functions in f exist such algorithms are of particular interest when f contains many irrelevant functions that should not appear in the aggregation procedure since selectors are suboptimal aggregation procedures this proves that two is the minimal number of elements of f required for the construction of an optimal aggregation procedure in every situations then we perform a numerical study for the problem of selection of the regularization parameters of the lasso and the elastic net estimators we compare on simulated examples our aggregation algorithms to aggregation with exponential weights to mallow s cp and to cross validation selection procedures

empirical processes and random projections in this note we establish some bounds on the supremum of certain empirical processes indexed by sets of functions with the same l2 norm we present several geometric applications of this result the most important of which is a sharpening of the johnson lindenstrauss embedding lemma our results apply to a large class of random matrices as we only require that the matrix entries have a subgaussian tail

the bm l _ mathbf 1 norm density estimator process the notion of an l1 norm density estimator process indexed by a class of kernels is introduced then a functional central limit theorem and a glivenko cantelli theorem are established for this process while assembling the necessary machinery to prove these results a body of poissonization techniques and restricted chaining methods is developed which is useful for studying weak convergence of general processes indexed by a class of functions none of the theorems imposes any condition at all on the underlying lebesgue density f also somewhat unexpectedly the distribution of the limiting gaussian process does not depend on f

robust partial shape classification using invariant breakpoints and dynamic alignment abstract a system for classifying partially occluded noisy shapes in varying positions orientations and dimensions is described an iterative algorithm derived from the colinearity principle is developed to locate invariant breakpoints on a shape contour the set of invariant breakpoints partitions the contour into a sequence of contour segments each contour segment is described by an ordered sequence that represents the euclidean distance between the pixels in the contour segment and the centroid of the region formed by connecting the end points of the contour segment by a straight line two stages of dynamic alignment with the appropriate constraints are formulated to determine shape similarity the first stage gives a distance measure of contour segment similarity and in the second stage the intersegment distances are used to optimally align contour segments so that a final measure of shape similarity is obtained the performance of the system is demonstrated by considering the classification of aircraft shapes belonging to four classes and classification results are presented as functions of noise and occlusion levels the results indicate that reasonable classification is obtained for noisy shapes with 0 to 30 occlusion

quasi perfect lee distance codes a construction of perfect quasiperfect lee distance codes in z sub k sup 2 is introduced for this class of codes a constant time encoding scheme is defined the minimum code distance is derived and the maximum covering radius is calculated efficient decoding schemes are investigated and developed in general a code of this class can be decoded in o t sub 1 where t sub 1 is the number of errors that can be corrected special cases however can be decoded in constant time

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

donsker classes of sets we study the central limit theorem clt and the law of large numbers lln for empirical processes indexed by a countable class of sets c the main result of purely measure theoretical nature relates different ways to measure the size of c it relies on a new rearrangement inequality that has been inspired by techniques used in the local theory of banach spaces as an application we give sharp necessary conditions for the clt that are in some sense the best possible we also obtain a way to compute the rate of convergence in the lln

an estimate on the supremum of a nice class of stochastic integrals and u statistics let a sequence of iid random variables 1 n be given on a space open image in new window with distribution together with a nice class open image in new window of functions f x 1 x k of k variables on the product space open image in new window for all f open image in new window we consider the random integral j n k f of the function f with respect to the k fold product of the normalized signed measure open image in new window where n denotes the empirical measure defined by the random variables 1 n and investigate the probabilities open image in new window for all x 0 we show that for nice classes of functions for instance if open image in new window is a vapnik cervonenkis class an almost as good bound can be given for these probabilities as in the case when only the random integral of one function is considered a similar result holds for degenerate u statistics too

on generic chaining and the smallest singular value of random matrices with heavy tails we present a very general chaining method which allows one to control the supremum of the empirical process suph h n 1 i 1nh2 xi eh2 in rather general situations we use this method to establish two main results first a quantitative non asymptotic version of the celebrated bai yin theorem on the singular values of a random matrix with i i d entries that have heavy tails and second a sharp estimate on the quadratic empirical process when h t t t t rn and is an isotropic unconditional log concave measure

kernel density estimators convergence in distribution for weighted sup norms let f n denote a kernel density estimator of a bounded continuous density f in the real line let t be a positive continuous function such that open image in new window under natural smoothness conditions necessary and sufficient conditions for the sequence open image in new window scriptstyle t inr big psi t fn t efn t big properly centered and normalized to converge in distribution to the double exponential law are obtained the proof is based on gaussian approximation and a new limit theorem for weighted sup norms of a stationary gaussian process this extends well known results of bickel and rosenblatt to the case of weighted sup norms with the sup taken over the whole line in addition all other possible limit distributions of the above sequence are identified subject to some regularity assumptions

the central limit theorem for stochastic processes the main result is that the necessary and sufficient conditions for the central limit theorem for centered second order processes given by gine and zinn 6 can be obtained without any basic measurability condition furthermore we extend some of their results

functional laws of the iterated logarithm for local empirical processes indexed by sets we introduce the notion of a multivariate local and tail empirical process indexed by sets and establish a number of functional laws of the iterated logarithm for such processes this leads to a unified methodology to study the almost sure behavior of various statistics which are local functionals of the empirical distribution such statistics include density estimators and the bahadur kiefer representation

weak convergence of the residual empirical process in explosive autoregression this paper proves the weak convergence of the residual empirical process in an explosive autoregression model to the brownian bridge as an application the kolmogorov smirnov goodness of fit test for testing that the errors have a specified distribution is shown to be asymptotically distribution free

perfect codes for metrics induced by circulant graphs an algebraic methodology for defining new metrics over two dimensional signal spaces is presented in this work we have mainly considered quadrature amplitude modulation qam constellations which have previously been modeled by quotient rings of gaussian integers the metric over these constellations based on the distance concept in circulant graphs is one of the main contributions of this work a detailed analysis of some degree four circulant graphs has allowed us to detail the weight distribution for these signal spaces a new family of perfect codes over gaussian integers will be defined and characterized by providing a solution to the perfect t dominating set problem over the circulant graphs presented finally we will show how this new metric can be extended to other signal sets by considering hexagonal constellations and circulant graphs of degree six

sequential complexities and uniform martingale laws of large numbers we establish necessary and sufficient conditions for a uniform martingale law of large numbers we extend the technique of symmetrization to the case of dependent random variables and provide sequential non i i d analogues of various classical measures of complexity such as covering numbers and combinatorial dimensions from empirical process theory we establish relationships between these various sequential complexity measures and show that they provide a tight control on the uniform convergence rates for empirical processes with dependent data as a direct application of our results we provide exponential inequalities for sums of martingale differences in banach spaces

aggregation via empirical risk minimization given a finite set f of estimators the problem of aggregation is to construct a new estimator whose risk is as close as possible to the risk of the best estimator in f it was conjectured that empirical minimization performed in the convex hull of f is an optimal aggregation method but we show that this conjecture is false despite that we prove that empirical minimization in the convex hull of a well chosen empirically determined subset of f is an optimal aggregation method

uniform limit theorems for harris recurrent markov chains we study uniform limit theorems for regenerative processes and get strong law of large numbers and central limit theorem of this type then we apply those results to harris recurrent markov chains based on some ideas of k athreya p ney and e nummelin

the law of large numbers for partial sum processes indexed by sets on donne des conditions necessaires et suffisantes pour la loi des grands nombres pour des processus de sommes partielles indicees par des ensembles de 0 1 d et basees sur des variables aleatoires integrables i i d

sample moduli for set indexed gaussian processes on etudie le comportement de la trajectoire de processus gaussiens wp indexes par des classes c de sous ensembles d un espace de probabilite x 7b a p

on a geometric notion of quantiles for multivariate data abstract an extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered the approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles both of which have been extensively studied in the literature these geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other l estimates of multivariate location and they lead to a directional notion of central and extreme points in a multidimensional setup such quantiles can be defined as meaningful and natural objects even in infinite dimensional hilbert and banach spaces and they yield an effective generalization of quantile regression in multiresponse linear model problems desirable equivariance properties are shown to hold for these multivariate quantiles and issues related to their computation for data in finite dimensional spaces are discussed n 1 2 consistenc

learning without concentration we obtain sharp bounds on the estimation error of the empirical risk minimization procedure performed in a convex class and with respect to the squared loss without assuming that class members and the target are bounded functions or have rapidly decaying tails rather than resorting to a concentration based argument the method used here relies on a small ball assumption and thus holds for classes consisting of heavy tailed functions and for heavy tailed targets the resulting estimates scale correctly with the noise level of the problem and when applied to the classical bounded scenario always improve the known bounds

on aggregation for heavy tailed classes we introduce an alternative to the notion of fast rate in learning theory which coincides with the optimal error rate when the given class happens to be convex and regular in some sense while it is well known that such a rate cannot always be attained by a learning procedure i e a procedure that selects a function in the given class we introduce an aggregation procedure that attains that rate under rather minimal assumptions for example that the l q and l 2 norms are equivalent on the linear span of the class for some q 2 and the target random variable is square integrable the key components in the proof include a two sided isomorphic estimator on distances between class members which is based on the median of means and an almost isometric lower bound of the form n 1 sum i 1 n f 2 x i ge 1 zeta mathbb e f 2 which holds uniformly in the class both results only require that the l q and l 2 norms are equivalent on the linear span of the class for some q 2

combining pac bayesian and generic chaining bounds there exist many different generalization error bounds in statistical learning theory each of these bounds contains an improvement over the others for certain situations or algorithms our goal is first to underline the links between these bounds and second to combine the different improvements into a single bound in particular we combine the pac bayes approach introduced by mcallester 1998 which is interesting for randomized predictions with the optimal union bound provided by the generic chaining technique developed by fernique and talagrand see talagrand 1996 in a way that also takes into account the variance of the combined functions we also show how this connects to rademacher based bounds

on rates of convergence and asymptotic normality in the multiknapsack problem in meanti et al 1990 an almost sure asymptotic characterization has been derived for the optimal solution value as function of the knapsack capacities when the profit and requirement coefficients of items to be selected from are random variables in this paper we establish a rate of convergence for this process using results from the theory of empirical processes

the central limit theorem for empirical processes under local conditions the case of radon infinitely divisible limits without gaussian component on obtient des resultats de convergence faible pour des processus empiriques indices par des classes f de fonctions dans le cas de limites de radon purement de poisson infiniment divisibles

bootstrapping general empirical measures it is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions mathscr f and based on a probability measure p holds a s if and only if mathscr f in mathrm clt p and int f 2 dp infty where f sup f in mathscr f f and it holds in probability if and only if mathscr f in mathrm clt p thus for a large class of statistics no local uniformity of the clt about p is needed for the bootstrap to work consistency of the bootstrap the bootstrapped law of large numbers is also characterized these results are proved under certain weak measurability assumptions on mathscr f

asymptotic score statistic processes and tests for constant hazard against a change point alternative the problem of testing for a constant failure rate against alternatives with failure rates involving a single change point is considered the asymptotic significance level for tests based on maximal score statistics are shown to involve the solution to a first passage time problem for an ornstein uhlenbeck process an example illustrates the methodology

on the law of the iterated logarithm for independent banach space valued random variables in this paper we establish some general forms of the law of the iterated logarithm for independent random variables xn with banach space values where xn is not necessarily identically distributed our results include the kolmogorov law of the iterated logarithm lil in both finite and infinite dimensional cases and they improve the wittmann lil as well as extend it to the vector setting the ledoux talagrand lil for an i i d sequence is also a simple corollary of our results

a clt for empirical processes involving time dependent data for stochastic processes xt t e we establish sufficient conditions for the empirical process based on ixt y pr xt y t e y r to satisfy the clt uniformly in t e y r corollaries of our main result include examples of classical processes where the clt holds and we also show that it fails for brownian motion tied down at zero and e 0 1

entropy and the combinatorial dimension we solve talagrand s entropy problem the l2 covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension this extends dudley s theorem on classes of 0 1 valued functions for which the shattering dimension is the vapnik chervonenkis dimension in convex geometry the solution means that the entropy of a convex body k is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of k this has a number of consequences including the optimal elton s theorem and estimates on the uniform central limit theorem in the real valued case

un critere sur les petites boules dans le theoreme limite central we show that a banach space valued random variablex such that mathop lim limits x to infty t 2 mathbb p left left x right t right 0 satisfies the central limit theorem if and only if the following criterion on small balls is fulfilled mathop lim limits x to infty t 2 mathbb p left left x right t right 0

quasi perfect codes from cayley graphs over integer rings the problem of searching for perfect codes has attracted great attention since the paper by golomb and welch in which the existence of these codes over lee metric spaces was considered since perfect codes are not very common the problem of searching for quasi perfect codes is also of great interest in this aspect also quasi perfect lee codes have been considered for 2 d and 3 d lee metric spaces in this paper constructive methods for obtaining quasi perfect codes over metric spaces modeled by means of gaussian and eisenstein jacobi integers are given the obtained codes form ideals of the integer ring thus preserving the property of being geometrically uniform codes moreover they are able to correct more error patterns than the perfect codes which may properly be used in asymmetric channels therefore the results in this paper complement the constructions of perfect codes previously done for the same integer rings finally decoding algorithms for the quasi perfect codes obtained in this paper are provided and the relationship of the codes and the lee metric ones is investigated

a non parametric entropy based approach to detect changes in climate extremes summary the paper focuses primarily on temperature extremes measured at 24 european stations with at least 90 years of data here the term extremes refers to rare excesses of daily maxima and minima as mean temperatures in this region have been warming over the last century it is automatic that this positive shift can be detected also in extremes after removing this warming trend we focus on the question of determining whether other changes are still detectable in such extreme events as we do not want to hypothesize any parametric form of such possible changes we propose a new non parametric estimator based on the kullback leibler divergence tailored for extreme events the properties of our estimator are studied theoretically and tested with a simulation study our approach is also applied to seasonal extremes of daily maxima and minima for our 24 selected stations

uniform bounds for norms of sums of independent random functions in this paper we develop a general machinery for finding explicit uniform probability and moment bounds on sub additive positive functionals of random processes using the developed general technique we derive uniform bounds on the ls norms of empirical and regression type processes usefulness of the obtained results is illustrated by application to the processes appearing in kernel density estimation and in nonparametric estimation of regression functions

graphs tessellations and perfect codes on flat tori quadrature amplitude modulation qam like signal sets are considered in this paper as coset constellations placed on regular graphs on surfaces known as flat tori such signal sets can be related to spherical block and trellis codes and may be viewed as geometrically uniform gu in the graph metric in a sense that extends the concept introduced by forney homogeneous signal sets of any order can then be labeled by a cyclic group induced by translations on the euclidean plane we construct classes of perfect codes on square graphs including lee spaces and on hexagonal and triangular graphs all on flat tori extension of this approach to higher dimensions is also considered

some central limit theorems for l valued semimartingales and their applications this paper is devoted to the generalization of central limit theorems for empirical processes to several types of l valued continuous time stochastic processes t x t n x t n where is a non empty set we deal with three kinds of situations as follows each coordinate process t x t n is i a general semimartingale ii a stochastic integral of a predictable function with respect to an integer valued random measure iii a continuous local martingale some applications to statistical inference problems are also presented we prove the functional asymptotic normality of generalized nelson aalen s estimator in the multiplicative intensity model for marked point processes its asymptotic efficiency in the sense of convolution theorem is also shown the asymptotic behavior of log likelihood ratio random fields of certain continuous semimartingales is derived

central limit theorems for stochastic processes under random entropy conditions necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row independent triangular array nj of stochastic processes indexed by a set s to a sample continuous gaussian process when the array satisfies a random entropy condition analogous to one used by gine and zinn 1984 for empirical processes this entropy condition is satisfied when s is a class of sets or functions with the vapnik ervonenkis property and each nj f fd nj is of the form njc for some reasonable random finite signed measure vnj as a result we obtain necessary and sufficient conditions for the weak convergence of possibly non i i d partial sum processes and new sufficient conditions for empirical processes indexed by vapnik ervonenkis classes special cases include prokhorov s 1956 central limit theorem for empirical processes and shorack s 1979 theorems on weighted empirical processes

lower bounds for the empirical minimization algorithm in this correspondence we present a simple argument that proves that under mild geometric assumptions on the class f and the set of target functions t the empirical minimization algorithm cannot yield a uniform error rate that is faster than 1 radic k in the function learning setup this result holds for various loss functionals and the target functions from t that cause the slow uniform error rate are clearly exhibited

the law of large numbers in locally convex spaces the law of large numbers is extended to random elements taking values in locally convex spaces the necessary and sufficient conditions for the law are given in a large class of locally convex spaces vix normed spaces this class includes among others the test function spaces and the distribution spaces

incoherent tensor norms and their applications in higher order tensor completion in this paper we investigate the sample size requirement for a general class of nuclear norm minimization methods for higher order tensor completion we introduce a class of tensor norms by allowing for different levels of coherence which allows us to leverage the incoherence of a tensor in particular we show that a k th order tensor of multilinear rank r and dimension d times cdots times d can be recovered perfectly from as few as o r k 1 2 d 3 2 r k 1 d log d 2 uniformly sampled entries through an appropriate incoherent nuclear norm minimization our results demonstrate some key differences between completing a matrix and a higher order tensor they not only point to potential room for improvement over the usual nuclear norm minimization but also highlight the importance of explicitly accounting for incoherence when dealing with higher order tensors although our focus is primarily on the theoretical guarantees for nuclear norm minimization such insights may prove useful for understanding performance of other related methods and developing improved practical algorithms

consistency for the least squares estimator in nonparametric regression we shall study the general regression model y g o x e where x and e are independent the available information about g 0 can be expressed by g 0 g for some class g as an estimator of g 0 we choose the least squares estimator we shall give necessary and sufficient conditions for consistency of this estimator in terms of basically geometric properties of g our main tool will be the theory of empirical processes

adaptive estimation in autoregression or mixing regression via model selection we study the problem of estimatingsome unknown regression function in a beta mixing dependent framework to this end we consider some collection of models which are finite dimensional spaces a penalized least squares estimator plse is built on a data driven selected model among this collection we state non asymptotic risk bounds for this plse and give several examples where the procedure can be applied autoregression regression with arithmetically beta mixing design points regression with mixing errors estimation in additive frameworks estimation of the order of the autoregression in addition we show that under a weak moment condition on the errors our estimator is adaptive in the minimax sense simultaneously over some family of besov balls

donsker classes and random geometry on donne des conditions geometriques aleatoires pour que le processus empirique indice par la classe des fonctions de carre integrable satisfasse le theoreme central limite

gaussian characterization of uniform donsker classes of functions it is proved that for classes of functions y satisfying some measurability the empirical processes indexed by f and based on p e 9 s satisfy the central limit theorem uniformly in p e 9w s if and only if the p brownian bridges gop indexed by y are sample bounded and pp uniformly continuous uniformly in p e 9 s uniform exponential bounds for empirical processes indexed by universal bounded donsker and uniform donsker classes of functions are also obtained

the central limit theorem and the law of iterated logarithm for empirical processes under local conditions a clt and a lil are proved under weak l2 gaussian bracketing conditions weaker than the usual ones these results have wide applicability and in particular provide an improvement of the jain marcus central limit theorem for c s valued random variables

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

rates of growth and sample moduli for weighted empirical processes indexed by sets probability inequalities are obtained for the supremum of a weighted empirical process indexed by a vapnik cervonenkis class c of sets these inequalities are particularly useful under the assumption p c c p c t 0 as t 0 they are used to obtain almost sure bounds on the rate of growth of the process as the sample size approaches infinity to find an asymptotic sample modulus for the unweighted empirical process and to study the ratio pn p of the empirical measure to the actual measure

geometrically uniform codes a signal space code c is defined as geometrically uniform if for any two code sequences in c there exists an isometry that maps one sequence into the other while leaving the code c invariant geometrical uniformity a strong kind of symmetry implies such properties as a the distance profiles from code sequences in c to all other code sequences are all the same and b all voronoi regions of code sequences in c have the same shape it is stronger than ungerboeck zehavi wolf symmetry or calderbank sloane regularity nonetheless most known good classes of signal space codes are shown to be generalized coset codes and therefore geometrically uniform including a lattice type trellis codes based on lattice partitions lambda lambda such that z sup n lambda lambda 4z sup n is a lattice partition chain and b phase shift keying psk type trellis codes based on up to four way partitions of a 2 sup n psk signal set

coupled samples in simulation assume that we wish to generate two samples of n independent identically distributed random variables x1 xn and y1 yn where x1 and y1 have densities f and g respectively if these samples are used in a simulation and f is close to g it is sometimes desirable to have close simulation results this can be achieved by insisting that both samples agree in most of their components that is xi yi for as many i as possible under the given distributional constraints samples with this property are said to be optimally coupled in this paper we propose and study various methods of coupling two samples a sequence of samples and an infinite family of samples

network and panel quantile effects via distribution regression abstract this paper provides a method to construct simultaneous confidence bands for quantile functions and quantile effects in nonlinear network and panel models with unobserved two way effects strictly exogenous covariates and possibly discrete outcome variables the method is based upon projection of simultaneous confidence bands for distribution functions constructed from fixed effects distribution regression estimators these fixed effects estimators are debiased to deal with the incidental parameter problem under asymptotic sequences where both dimensions of the data set grow at the same rate the confidence bands for the quantile functions and effects have correct joint coverage in large samples an empirical application to gravity models of trade illustrates the applicability of the methods to network data

perfect codes from cayley graphs over lipschitz integers the search for perfect error correcting codes has received intense interest since the seminal work by hamming decades ago golomb and welch studied perfect codes for the lee metric in multidimensional torus constellations in this work we focus our attention on a new class of four dimensional signal spaces which include tori as subcases our constellations are modeled by means of cayley graphs defined over quotient rings of lipschitz integers previously unexplored perfect codes of length one will be provided in a constructive way by solving a typical problem of vertices domination in graph theory the codewords of such perfect codes are constituted by the elements of a principal left ideal of the considered quotient ring the generalization of these techniques for higher dimensional spaces is also considered in this work by modeling their signal sets through cayley dickson algebras

on weakly bounded empirical processes let f be a class of functions on a probability space and let x 1 x k be independent random variables distributed according to we establish an upper bound that holds with high probability on rm sup f in f i f x i geq t for every t 0 and that depends on a natural geometric parameter associated with f we use this result to analyze the supremum of empirical processes of the form z f left k 1 sum i 1 k f p x i mathbb e f p right for p 1 using the geometry of f we also present some geometric applications of this approach based on properties of the random operator gamma k 1 2 sum i 1 k x i e i where x i i 1 k are sampled according to an isotropic log concave measure on mathbb r n

high frequency donsker theorems for levy measures donsker type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate levy process in the asymptotic regime the sampling frequencies increase to infinity and the limiting object is a gaussian process that can be obtained from the composition of a brownian motion with a covariance operator determined by the levy measure the results are applied to derive the asymptotic distribution of natural estimators for the distribution function of the levy jump measure as an application we deduce kolmogorov smirnov type tests and confidence bands

the law of the iterated logarithm in uniformly convex banach spaces we give necessary and sufficient conditions for a random vari able x with values in a uniformly convex banach space b to satisfy the law of the iterated logarithm precisely we show that a b valued random vari able x satisfies the compact law of the iterated logarithm if and only if a 2 i2 x oo the family i x 2 x 6 b i 1 is uni formly integrable and sn a n 0 in probability 1 introduction let b be a banach space with topological dual b and norm by random variable with values in b we mean a measurable map x from some probability space 0 7 p into b such that the image of p by x defines a radon probability measure on the borel cr field of b we may assume equivalently that x takes almost surely its values in a separable subspace of b given a b valued random variable x we denote by xn new a sequence of inde pendent copies of x and let as usual sn xx h h xn we write lt to denote the function max l logi and use l2t to denote l lt t g r we set further or 2nl2n 1 2 the random variable x is said to satisfy the bounded law of the iterated logarithm lil if the sequence sn an n s is bounded with probability one and the compact lil if this sequence is almost surely relatively compact in b it is well known that if x satisfies the compact lil there is a compact convex and symmetric set k in b such that

ranking and empirical minimization of u statistics the problem of ranking ordering instances instead of simply classifying them has recently gained much attention in machine learning in this paper we formulate the ranking problem in a rigorous statistical framework the goal is to learn a ranking rule for deciding among two instances which one is better with minimum ranking risk since the natural estimates of the risk are of the form of a u statistic results of the theory of u processes are required for investigating the consistency of empirical risk minimizers we establish in particular a tail inequality for degenerate u processes and apply it for showing that fast rates of convergence may be achieved under specific noise assumptions just like in classification convex risk minimization methods are also studied

modeling toroidal networks with the gaussian integers in this paper we consider a broad family of toroidal networks denoted as gaussian networks which include many previously proposed and used topologies we will define such networks by means of the gaussian integers the subset of the complex numbers with integer real and imaginary parts nodes in gaussian networks are labeled by gaussian integers which confer these topologies an algebraic structure based on quotient rings of the gaussian integers in this sense gaussian integers reveal themselves as the appropriate tool for analyzing and exploiting any type of toroidal network using this algebraic approach we can characterize the main distance related properties of gaussian networks providing closed expressions for their diameter and average distance in addition we solve some important applications like unicast and broadcast packet routing or the perfect placement of resources over these networks