existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations in this paper the banach fixed point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference email protected n dx n b n x email protected email protected n x h 1 n x h 2 n x h k n f n x f 1 n x f 2 n x f k n c n n n 0 where t email protected n n email protected n 0 a b c n n 0 r with a n 0 for email protected n n 0 h f n n 0xr k r and h l f l n n 0 z withlimn h l n limn f l n email protected 1 2 k a few mann type iterative approximation schemes with errors are suggested and the errors estimates between the iterative approximations and the nonoscillatory solutions are discussed seven nontrivial examples are given to illustrate the advantages of our results

differentiability of t functionals of location and scatter the paper aims at finding widely and smoothly defined nonparametric location and scatter functionals as a convenient vehicle maximum likelihood estimation of the location vector and scatter matrix e of an elliptically symmetric t distribution on r d with degrees of freedom v i extends to an m functional defined on all probability distributions p in a weakly open weakly dense domain u here u consists of p putting not too much mass in hyperplanes of dimension d as shown for empirical measures by kent and tyler ann statist 19 1991 2102 2119 it will be seen here that e is analytic on u for the bounded lipschitz norm or for d 1 for the sup norm on distribution functions for k 1 2 and other norms depending on k and more directly adapted to t functionals one has continuous differentiability of order k allowing the delta method to be applied to for any p in u which can be arbitrarily heavy tailed these results imply asymptotic normality of the corresponding m estimators n n in dimension d 1 only the t v functional a extends to be defined and weakly continuous at all p

uniform central limit theorems for kernel density estimators let mathbb p n ast k h n x n 1 h n d sum i 1 n k left x x i h n right be the classical kernel density estimator based on a kernel k and n independent random vectors x i each distributed according to an absolutely continuous law mathbb p on mathbb r d it is shown that the processes f longmapsto sqrt n int fd mathbb p n ast k h n mathbb p f in mathcal f converge in law in the banach space ell infty mathcal f for many interesting classes mathcal f of functions or sets some mathbb p donsker some just mathbb p pregaussian the conditions allow for the classical bandwidths h n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error thus showing that subject to some natural conditions kernel density estimators are plug in estimators in the sense of bickel and ritov ann statist 31 1033 1053 2003 some new results on the uniform central limit theorem for smoothed empirical processes needed in the proofs are also included

uniformly semiparametric efficient estimation of treatment effects with a continuous treatment this article studies identification estimation and inference of general unconditional treatment effects models with continuous treatment under the ignorability assumption we show identification of the parameters of interest the dose response functions under the assumption that selection to treatment is based on observables we propose a semiparametric two step estimator and consider estimation of the dose response functions through moment restriction models with generalized residual functions that are possibly nonsmooth this general formulation includes average and quantile treatment effects as special cases the asymptotic properties of the estimator are derived namely uniform consistency weak convergence and semiparametric efficiency we also develop statistical inference procedures and establish the validity of a bootstrap approach to implement these methods in practice monte carlo simulations show that the proposed methods have good finite sample properties finally we apply the proposed

stability properties of empirical risk minimization over donsker classes we study some stability properties of algorithms which minimize or almost minimize empirical error over donsker classes of functions we show that as the number n of samples grows the l2 diameter of the set of almost minimizers of empirical error with tolerance n o n 1 2 converges to zero in probability hence even in the case of multiple minimizers of expected error as n increases it becomes less and less likely that adding a sample or a number of samples to the training set will result in a large jump to a new hypothesis moreover under some assumptions on the entropy of the class along with an assumption of komlos major tusnady type we derive a power rate of decay for the diameter of almost minimizers this rate through an application of a uniform ratio limit inequality is shown to govern the closeness of the expected errors of the almost minimizers in fact under the above assumptions the expected errors of almost minimizers become closer with a rate strictly faster than n 1 2

oscillation of second order nonlinear difference equation with continuous variable in this paper we are mainly concerned with the second order nonlinear difference equation with continuous variable here by using the iterated integral transformations generalized riccati transformations and integrating factors we give some oscillatory criteria for this equation

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

inference on counterfactual distributions counterfactual distributions are important ingredients for policy analysis and decomposition analysis in empirical economics in this article we develop modeling and inference tools for counterfactual distributions based on regression methods the counterfactual scenarios that we consider consist of ceteris paribus changes in either the distribution of covariates related to the outcome of interest or the conditional distribution of the outcome given covariates for either of these scenarios we derive joint functional central limit theorems and bootstrap validity results for regression based estimators of the status quo and counterfactual outcome distributions these results allow us to construct simultaneous confidence sets for function valued effects of the counterfactual changes including the effects on the entire distribution and quantile functions of the outcome as well as on related functionals these confidence sets can be used to test functional hypotheses such as no effect positive effect or stochastic dominance our theory applies to general counterfactual changes and covers the main regression methods including classical quantile duration and distribution regressions we illustrate the results with an empirical application to wage decompositions using data for the united states as a part of developing the main results we introduce distribution regression as a comprehensive and flexible tool for modeling and estimating the entire conditional distribution we show that distribution regression encompasses the cox duration regression and represents a useful alternative to quantile regression we establish functional central limit theorems and bootstrap validity results for the empirical distribution regression process and various related functionals publication abstract

a donsker theorem for levy measures abstract given n equidistant realisations of a levy process l t t 0 a natural estimator n n for the distribution function n of the levy measure is constructed under a polynomial decay restriction on the characteristic function a donsker type theorem is proved that is a functional central limit theorem for the process n n n n in the space of bounded functions away from zero the limit distribution is a generalised brownian bridge process with bounded and continuous sample paths whose covariance structure depends on the fourier integral operator f 1 1 the class of levy processes covered includes several relevant examples such as compound poisson gamma and self decomposable processes main ideas in the proof include establishing pseudo locality of the fourier integral operator and recent techniques from smoothed empirical processes

donsker type theorems for nonparametric maximum likelihood estimators let mathcal p be a nonparametric probability model consisting of smooth probability densities and let hat p n be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law mathbb p with hat mathbb p n denoting the measure induced by the density hat p n define the stochastic process hat nu n f longmapsto sqrt n int fd hat mathbb p n mathbb p where f ranges over some function class mathcal f we give a general condition for donsker classes mathcal f implying that the stochastic process hat nu n is asymptotically equivalent to the empirical process in the space ell infty mathcal f of bounded functions on mathcal f this implies in particular that hat nu n converges in law in ell infty mathcal f to a mean zero gaussian process we verify the general condition for a large family of donsker classes mathcal f we give a number of applications convergence of the probability measure hat mathbb p n to mathbb p at rate sqrt n in certain metrics metrizing the topology of weak star convergence a unified treatment of convergence rates of the mle in a continuous scale of sobolev norms sqrt n efficient estimation of nonlinear functionals defined on mathcal p limit theorems at rate sqrt n for the maximum likelihood estimator of the convolution product mathbb p ast p

stochastic estimation and testing on introduit plusieurs procedures stochastiques pour lesquelles on developpe une theorie asymptotique estimations et tests de distance minimum stochastique estimation du maximum de vraisemblance stochastique ensembles de confiance pour une probabilite inconnue etc

concentration inequalities and asymptotic results for ratio type empirical processes let be a class of measurable functions on a measurable space s mathcal s with values in 0 1 and let pn n 1 i 1n xi be the empirical measure based on an i i d sample x1 xn from a probability distribution p on s mathcal s we study the behavior of suprema of the following type sup r n textless sigma p f leq delta n frac p n f pf phi sigma p f where pf var1 2pf and is a continuous strictly increasing function with 0 0 using talagrand s concentration inequality for empirical processes we establish concentration inequalities for such suprema and use them to derive several results about their asymptotic behavior expressing the conditions in terms of expectations of localized suprema of empirical processes we also prove new bounds for expected values of sup norms of empirical processes in terms of the largest pf and the l2 p norm of the envelope of the function class which are especially suited for estimating localized suprema with this technique we extend to function classes most of the known results on ratio type suprema of empirical processes including some of alexander s results for vc classes of sets we also consider applications of these results to several important problems in nonparametric statistics and in learning theory including general excess risk bounds in empirical risk minimization and their versions for l2 regression and classification and ratio type bounds for margin distributions in classification

relationships between donsker classes and sobolev spaces let h be a separable real hubert space let x be a second countable topological space and x p be a probability space where is the borel sets let t h c b x be linear and continuous then the image of the unit ball of h is a donsker class so if k 1 l 2 then the unit ball of the sobolev space f in l 2 mathbb r n smallint hat fk 2 tfrac 1 2 infty is a donsker class for any p for most other k it is not a donsker class for any p with a bounded density

global solvability for a second order nonlinear neutral delay difference equation this paper studies the global existence of solutions of the second order nonlinear neutral delay difference equation d a email protected x n bx n t f n x n d 1 n x n d 2 n x n d k n c n n n 0 with respect to all email protected r a few results on global existence of uncountably many bounded nonoscillatory solutions are established for the above difference equation several nontrivial examples which dwell upon the importance of the results obtained in this paper are also included

model averaging in semiparametric estimation of treatment effects choosing the covariates and functional form of the propensity score is an important choice for estimating treatment effects this paper proposes a data driven way of averaging the estimators over candidate specifications to resolve the specification uncertainty in the propensity score weighting estimation of the att the proposed procedures minimize the estimated mse of the att estimator in a local asymptotic framework we formulate model averaging as a statistical decision problem in a limit experiment and derive an averaging scheme that is bayes optimal with respect to a given prior the averaging estimator outperforms selection estimators and the estimators in any of the candidate models in terms of bayes asymptotic mse our monte carlo studies illustrate the size of the mse gains we apply the averaging procedure to evaluate the effect of a labor market program

frechet differentiability p variation and uniform donsker classes classically a stastistical functional is defined on a space of distribution f on the real line with the supremum norn the values may be real or themselves functions such as the quantile function f 1 nonlinear functionals are studied via their derivatives this paper and a related one dudley 1991a will show that the differentiability properties originally proved by reeds 1976 also treated in fernholz 1983 can be improved substantially

global sensitivity analysis for repeated measures studies with informative drop out a semi parametric approach in practice both testable and untestable assumptions are generally required to draw inference about the mean outcome measured at the final scheduled visit in a repeated measures study with drop out scharfstein et al 2014 proposed a sensitivity analysis methodology to determine the robustness of conclusions within a class of untestable assumptions in their approach the untestable and testable assumptions were guaranteed to be compatible their testable assumptions were based on a fully parametric model for the distribution of the observable data while convenient these parametric assumptions have proven especially restrictive in empirical research here we relax their distributional assumptions and provide a more flexible semi parametric approach we illustrate our proposal in the context of a randomized trial for evaluating a treatment of schizoaffective disorder

oscillation criteria for second order functional difference equations with summation small coefficient we consider the second order functional difference equation a n n p n x g n 0 where n p n are sequences of integers with a n 0 0 g n n limn g n p n 0 and p n 0 we obtain some necessary conditions for equation 1 exists nonoscillatory solutions and some sufficient conditions for equation 1 is oscillatory

bounded oscillation criteria for certain third order nonlinear difference equations with several delays and advances this paper deals with the third order nonlinear difference equations with several delays and advances d a email protected 2x n f n x n x n d 1 n x n d k n 0 n n 0 and d a email protected 2x n b ng x n x n d 1 n x n d k n 0 n n 0 sufficient conditions of the existence of uncountably many bounded positive negative and nonoscillatory solutions respectively and necessary and sufficient conditions of the bounded oscillation of and with certain asymptotic behavior are given in addition a sufficient condition of the oscillatory behavior of all bounded solutions of with a special of asymptotic property is also presented nine nontrivial examples which dwell upon the importance of the results obtained in this paper are also included

empirical processes indexed by lipschitz functions on obtient des conditions necessaires et suffisantes sur p pour que les boules unites de bl r et lip r soient des classes de p donsker fonctionnelles

periodic solutions of duffing s equations with superquadratic potential abstract this paper is devoted to the study of harmonic and subharmonic solutions for the second order scalar nonlinear duffing s equation x g x p t x x where g and p are continuous functions with p bounded and periodic in the first variable and g satisfying the assumption g x sign x as x among other results we prove the existence of infinitely many harmonic and subharmonic solutions of any order p p t and if the potential g x of g x satisfies certain conditions of superquadratic growth at the new existence results can be applied to situations in which the more classical superlinear growth condition g x x as x is not satisfied in this manner various preceding theorems are improved and sharpened see the introduction for more details proofs are based on a generalized version of the poincare birkhoff twist theorem due to w ding

linear oscillation of second order nonlinear difference equations with damped term abstract in this paper we shall establish relations between the oscillation of second order nonlinear difference equations with damped term and the oscillation of their linear limiting equations

a second order nonlinear difference equation oscillation and asymptotic behavior discrete analogues are investigated for well known results on oscillation growth and asymptotic behavior of solutions of y q t y 0 for q t 0 and for q t 0 the analogue of atkinson s oscillation criterion is shown to be true for 2yn 1 qn yn 0 but the analogue for atkinson s nonoscillation criterion is shown to be false

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

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

efficient nonparametric inference for discretely observed compound poisson processes a compound poisson process whose parameters are all unknown is observed at finitely many equispaced times nonparametric estimators of the jump and levy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions the limiting gaussian processes are identified and efficiency of the estimators is established kernel estimators for the mass function the intensity and the drift are also proposed their asymptotic properties including efficiency are analysed and joint asymptotic normality is shown inference tools such as confidence regions and tests are briefly discussed

metric marginal problems for set valued or non measurable variables in a separable metric space if two borel probability measures laws are nearby in a suitable metric then there exist random variables with those laws which are nearby in probability specifically by a well known theorem of strassen the prohorov distance between two laws is the infimum of ky fan distances of random variables with those laws the present paper considers possible extensions of strassen s theorem to two random elements one of which may be compact set valued and or non measurable there are positive results in finite dimensional spaces but with factors depending on the dimension examples show that such factors cannot entirely be avoided so that the extension of strassen s theorem to the present situation fails in infinite dimensions

nonparametric estimators which can be plugged in we consider nonparametric estimation of an object such as a probability density or a regression function can such an estimator achieve the ratewise minimax rate of convergence on suitable function spaces while at the same time when plugged in estimate efficiently at a rate of n 1 2 with the best constant many functionals of the object for example can we have a density estimator whose definite integrals are efficient estimators of the cumulative distribution function we show that this is impossible for very large sets for example expectations of all functions bounded by m however we also show that it is possible for sets as large as indicators of all quadrants that is distribution functions we give appropriate constructions of such estimates

oscillation theorems for second order advanced functional difference equations abstract we consider the second order advanced functional difference equation a n n p n g n 0 where a n 0 s n 0 1 a s p n 0 p n 0 g n n 1 g n is a monotone increasing integer sequence we obtain some new oscillation criteria through an appropriate riccati equation

oscillation results for second order scalar and matrix difference equations we consider the self adjoint second order scalar difference equation rn xn pnxn 1 0 1 and the matrix system rn xn pnxn 1 0 2 where rn 0 pn 0 rn 0 pn 0 are sequences of real numbers d d hermitian matrices with rn 0 rn 0 we obtain some new oscillation criteria through change of variable techniques combined with consideration of 1 and 2 as perturbations of the cases pn 0 pn 0

universal donsker classes and metric entropy when mathfrak f is a universal donsker class then for independent indetically distributed i i d observation mathbf x 1 ldots mathbf x n with an unknown law p for any mathfrak f i in mathfrak f i 1 ldots m quad n 1 2 left mathfrak f 1 left mathbf x 1 right ldots mathfrak f i left mathbf x n right right 1 leq i leq m is asymptotically normal with mean vector n 1 2 left int mathfrak f i left mathbf x n right d mathbf p left x right right 1 leq i leq m and covariance matrix int mathfrak f i mathfrak f j d mathbf p int mathfrak f id mathbf p int mathfrak f jd mathbf p uniformly for mathfrak f i in mathfrak f then for certain statistics formed frome the mathfrak f i left mathbf x k right even where mathfrak f i may be chosen depending on the mathbf x k there will be asymptotic distribution as n rightarrow infty for example for mathbf x 2 statistics where f i are indicators of disjoint intervals depending suitably on mathbf x 1 ldots mathbf x n whose union is the real line mathbf x 2 quadratic forms have limiting distributions roy 1956 and watson 1958 which may however not be mathbf x 2 distributions and may depend on p chernoff and lehmann 1954 universal donsker classes of sets are up to mild measurability conditions just classes satisfying the vapnik cervonenkis comdinatorial conditions defined later in this section donsker the vapnik cervonenkis combinatorial conditions defined later in this section durst and dudley 1981 and dudley 1984 chapter 11 the use of such classes allows a variety of extensions of the roy watson results to general multidimensional sample spaces pollard 1979 and moore and subblebine 1981 vapnik and cervonenkis 1974 indicated application of their families of sets to classification pattern recognition problems more recently the classes have been applied to tree structured classifiacation breiman friedman olshen and stone 1984 chapter 12

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

using a monotone single index model to stabilize the propensity score in missing data problems and causal inference the augmented inverse weighting method is one of the most popular methods for estimating the mean of the response in causal inference and missing data problems an important component of this method is the propensity score popular parametric models for the propensity score include the logistic probit and complementary log log models a common feature of these models is that the propensity score is a monotonic function of a linear combination of the explanatory variables to avoid the need to choose a model we model the propensity score via a semiparametric single index model in which the score is an unknown monotonic nondecreasing function of the given single index under this new model the augmented inverse weighting estimator aiwe of the mean of the response is asymptotically linear semiparametrically efficient and more robust than existing estimators moreover we have made a surprising observation the inverse probability weighting and aiwes based on a correctly specified parametric model may have worse performance than their counterparts based on a nonparametric model a heuristic explanation of this phenomenon is provided a real data example is used to illustrate the proposed methods

combinatorics of random processes and sections of convex bodies we find a sharp combinatorial bound for the metric entropy of sets in r n and general classes of functions this solves two basic combinatorial conjectures on the empirical processes 1 a class of functions satisfies the uniform central limit theorem if the square root of its combinatorial dimension is integrable 2 the uniform entropy is equivalent to the combinatorial dimension under minimal regularity our method also constructs a nicely bounded coordinate section of a symmetric convex body in r n in the operator theory this essentially proves for all normed spaces the restricted invertibility principle of bourgain and tzafriri

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

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

doubly robust inference for targeted minimum loss based estimation in randomized trials with missing outcome data missing outcome data is a crucial threat to the validity of treatment effect estimates from randomized trials the outcome distributions of participants with missing and observed data are often different which increases bias causal inference methods may aid in reducing the bias and improving efficiency by incorporating baseline variables into the analysis in particular doubly robust estimators incorporate 2 nuisance parameters the outcome regression and the missingness mechanism ie the probability of missingness conditional on treatment assignment and baseline variables to adjust for differences in the observed and unobserved groups that can be explained by observed covariates to consistently estimate the treatment effect one of these nuisance parameters must be consistently estimated traditionally nuisance parameters are estimated using parametric models which often precludes consistency particularly in moderate to high dimensions recent research on missing data has focused on data adaptive estimation to help achieve consistency but the large sample properties of such methods are poorly understood in this article we discuss a doubly robust estimator that is consistent and asymptotically normal under data adaptive estimation of the nuisance parameters we provide a formula for an asymptotically exact confidence interval under minimal assumptions we show that our proposed estimator has smaller finite sample bias compared to standard doubly robust estimators we present a simulation study demonstrating the enhanced performance of our estimators in terms of bias efficiency and coverage of the confidence intervals we present the results of an illustrative example a randomized double blind phase 2 3 trial of antiretroviral therapy in hiv infected persons

the shattering dimension of sets of linear functionals we evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets the proofs here relay mostly on methods from the local theory of normed spaces and include volume estimates factorization techniques and tail estimates of norms viewed as random variables on euclidean spheres the estimates of shattering dimensions can be applied to obtain error bounds for certain classes of functions a fact which was the original motivation of this study although this can probably be done in a more traditional manner we also use the approach presented here to determine whether several classes of linear functionals satisfy the uniform law of large numbers and the uniform central limit theorem

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

oscillation of solutions for second order nonlinear difference equations with nonlinear neutral term abstract some riccati type difference inequalities are given for the second order nonlinear difference equations with nonlinear neutral term and using these inequalities we obtain some oscillation criteria for the above equation

an improved bootstrap test of stochastic dominance we propose a new method of testing stochastic dominance that improves on existing tests based on the standard bootstrap or subsampling the method admits prospects involving infinite as well as finite dimensional unknown parameters so that the variables are allowed to be residuals from nonparametric and semiparametric models the proposed bootstrap tests have asymptotic sizes that are less than or equal to the nominal level uniformly over probabilities in the null hypothesis under regularity conditions this paper also characterizes the set of probabilities so that the asymptotic size is exactly equal to the nominal level uniformly as our simulation results show these characteristics of our tests lead to an improved power property in general the improvement stems from the design of the bootstrap test whose limiting behavior mimics the discontinuity of the original test s limiting distribution

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

vapnik chervonenkis type conditions and uniform donsker classes of functions given a bounded class of functions we introduce a combinatorial quantity related to the idea of vapnik chervonenkis classes that is much more explicit than the koltchinskii pollard entropy but is proved to be essentially of the same order

invariance principles for sums of banach space valued random elements and empirical processes almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable banach space it is then shown that empirical processes readily fit into this general framework thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes