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

empirical likelihood based confidence interval for the mean with a heavy tailed distribution empirical likelihood based confidence intervals for a mean were introduced by owen biometrika 75 1988 237 249 where at least a finite second moment is required this excludes some important distributions for example those in the domain of attraction of a stable law with index between 1 and 2 in this article we use a method similar to qin and wong scand j statist 23 1996 209 219 to derive an empirical likelihood based confidence interval for the mean when the underlying distribution has heavy tails our method can easily be extended to obtain a confidence interval for any order of moment of a heavy tailed distribution

on the asymptotic expansion of the empirical process of long memory moving averages let x n i 1 a i e n i where the e i are iid with mean 0 and finite fourth moment and the a i are regularly varying with index where 1 2 1 so that x n has long range dependence this covers an important class of the fractional arima process for r 0 let y n r n n 1 1 j1 jr r s 1 a js e n js y n 0 n 2 n r var y n r and f r the rth derivative of the distribution function of x n the y n r are uncorrelated and are stochastically decreasing in r for any positive integer p 2 1 1 it is shown under mild regularity conditions that with probability 1 n n 1 i x n x p r 0 1 r f r x y n r o n n p uniformly for all x r o 1 2 1 2 p 1 2 this generalizes a host of existing results and provides the vehicle for a number of statistical applications

von neumann stability analysis of first order accurate discretization schemes for one dimensional 1d and two dimensional 2d fluid flow equations abstract in literature the von neumann stability analysis of simplified model equations such as the wave equation is typically used to determine stability conditions for the non linear partial differential fluid flow equations navier stokes and euler however practical experience suggests that such simplistic stability conditions are grossly inadequate for computations involving the system of coupled flow equations the goal of this paper is to determine stability conditions for the full system of fluid flow equations the euler equations are examined as any conditions derived for the euler equations will apply to the navier stokes ns equations in the limit of convection dominated flows a von neumann stability analysis is conducted for the one dimensional 1d and two dimensional 2d euler equations the system of equations is discretized on a staggered grid using finite difference discretization techniques the use of a staggered grid allows equivalence to finite volume discretization by combining the different discretization techniques ten solution schemes are formulated eight solution schemes are considered for the 1d euler equations and two schemes for the 2d euler equations for each scheme error amplification matrices are determined from the stability analysis stable and unstable regimes are identified and practical stability limits are predicted in terms of the maximum allowable cfl courant friedrichs lewy number as a function of mach number the predictions are verified for selected schemes using the riemann problem at incompressible and compressible mach numbers very good agreement is obtained between the analytically predicted and the experimentally observed cfl values the successfully tested stability limits are presented in graphical form which offer a viable alternative to complicated mathematical expressions often reported in published literature and should benefit everyday cfd computational fluid dynamics users the stability regions are used to discuss the effect of time integration explicit vs implicit density bias in continuity equation and momentum convection term linearization on stability a comparison of the predicted stability limits for 1d and 2d euler equations with commonly used stability conditions arising from the wave equation shows that the stability thresholds for the euler equations lie well below those predicted by the wave equation analysis in addition the 2d euler stability limits are more restrictive as compared to 1d euler limits since the present analysis accounts for the full system of fluid flow euler equations the derived stability conditions can be used by cfd practitioners to estimate a timestep or cfl number to guide the stability of their computations

on a problem concerning spacings let un n 1 2 be a sequence of independent uniformly distributed random variables on 0 1 and si n i 1 n be the spacings induced by the order statistics of u1 un 1 denote begin array l m l n rm max min s i j n rm 1 mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x i mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x n l 1 rm 0 mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x j l end array the exact distribution of m2 n is determined and based on analysis of something called fibonacci distribution the weak and almost sure convergence of the sequence m2 n are discussed furthermore the limiting distribution of ml n is determined for any fixed l and the equation p mathop lim limits n to infty rm nl n m l n n log n 1 1 is shown for some sequence ln

strassen type functional laws for strong topologies letw denote a wiener process the functional law of the iterated logarithm due to strassen 1964 establishes that the sequence 2nlln 1 2w ns of functions ofs 0 1 is almost surely compact in the uniform topology and gives a simple description of the corresponding limit set in this paper we obtain a general characterization of the topologies under which this statement remains valid

a glivenko cantelli lemma and weak convergence for empirical processes of associated sequences with the help of an extension of hoeffding s equality we develop a way for estimating the covariance structures for empirical functions of associated sequences in terms of covariances of the original random variables based on these estimations a glivenko cantelli lemma for associated sequences and weak convergence for empirical processes of stationary associated sequences are obtained all under the conditions on covariances of the original random variables

convergence rates for the bootstrapped product limit process on etablit des taux d approximation forte pour le processus limite produit du bootstrap et le processus quantile correspondant

upper and lower class sequences for minimal uniform spacings letkn be thek th smallest spacing defined byx1 xn independent random variables uniformly distributed on 0 1 wherek 1 is a fixed integer letun be a sequence of positive numbers satisfyingun n2 we show that p n 2 k n leqq u n i o 0 1 when sum limits n u n k n infty infty i begin gathered if in addition u n uparrow then p n 2 k n geqq u n i o 0 1 when hfill sum limits n u n k n exp u n infty infty hfill end gathered ii

the a s behavior of the weighted empirical process and the lil for the weighted tail empirical process the tail empirical process is defined to be for each n n wn t n kn 1 2an tkn n 0 t 1 where an is the empirical process based on the first n of a sequence of independent uniform 0 1 random variables and kn 8 n 1 is a sequence of positive numbers with kn n 0 and kn 8 in this paper a complete description of the almost sure behavior of the weighted empirical process anan q where q is a weight function and an 8 n 1 is a sequence of positive numbers is established as well as a characterization of the law of the iterated logarithm behavior of the weighted tail empirical process wn q provided kn loglog n 8 these results unify and generalize several results in the literature moreover a characterization of the central limit theorem behavior of wn q is presented that result is applied to the construction of asymptotic confidence bands for intermediate quantiles from an arbitrary continuous distribution

what portion of the sample makes a partial sum asymptotically stable or normal let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0 2 be given we show that if at each stage n a number k n depending on n of the lower and upper order statistics are removed from the n th partial sum of the given random variables then under appropriate conditions on k n the remaining sum can be normalized to converge in distribution to a standard normal random variable a further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour our main tool is a new brownian bridge approximation to the uniform empirical process in weighted supremum norms

normal and stable convergence of integral functions of the empirical distribution function we prove general invariance principles for integral functions of the empirical process as corollaries we derive probabilistic proofs of the sufficiency criteria for a distribution to belong to the domain of attraction of the normal and stable laws with index 0 a 2 in the process we obtain equivalent statements of these criteria in terms of the tail behaviour of the underlying quantile function we also give a representation of any stable random variable with index 0 a 2 in terms of a linear combination of two independent and identically distributed poisson integrals the role of a fixed number of extreme terms is exactly determined

on the bahadur representation of sample quantiles for dependent sequences we establish the bahadur representation of sample quantiles for linear and some widely used nonlinear processes local fluctuations of empirical processes are discussed applications to the trimmed and winsorized means are given our results extend previous ones by establishing sharper bounds under milder conditions and thus provide new insight into the theory of empirical processes for dependent random variables

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

strong approximations of some biometric estimates under random censorship the empirical distribution function has been widely used as an estimator for the distribution function of the elements of a random sample it is not however appropriate when the observations are incomplete this is the case for example when the observations have been randomly censored on the right in the latter case the so called product limit estimate of kaplan and meier 16 has been generally accepted as a substitute for the empirical distribution function and it possesses many of the same properties cf meier 19 developing the corresponding statement of efron 8 weak convergence over a finite interval of the kaplan meier product limit estimate to a gaussian process when there is censorship on the right has been obtained by breslow and crowley 3 cf also meier 19 breslow and crowley obtained their result via the limiting distribution of the closely related cumulative hazard process under random censorship a generalisation of the efron breslow crowley theorem was formulated by yang 24 for the case when censorship on the right is performed by more competing risks these results have numerous statistical applications in areas such as medical follow up studies life testing actuarial sciences and demography koziol and green 18 have computed tables for the limiting distribution of the cram6r von mises statistic based on the kaplan meier product limit estimate under a specific random censorship model a point will be stressed in this paper however that by a certain transformation which seems to have escaped many researchers attention and is due to efron 8 it is always possible to obtain the standard wiener process in the limit the distributions of many functionals of which are well known this transformation was only recently used in a stimulating note by gillespie and fisher 14 to construct asymptotic confidence bands for the unknown distribution function in the kaplan meier model a general scheme of random censorship can be defined in the following way let x be a real random variable with distribution function f t

strong laws for local quantile processes we show that increments of size h n from the uniform quantile and uniform empirical processes in the neighborhood of a fixed point t 0 0 1 may have different rates of almost sure convergence to 0 in the range where h n 0 and nh n log n in particular when h n n with 0 0 the best possible uniform almost sure rate of approximation of the uniform quantile process by a normed kiefer process is not better than o n 1 4 log n e

innovated higher criticism for detecting sparse signals in correlated noise higher criticism is a method for detecting signals that are both sparse and weak although first proposed in cases where the noise variables are independent higher criticism also has reasonable performance in settings where those variables are correlated in this paper we show that by exploiting the nature of the correlation performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer indeed it turns out that the case of independent noise is the most difficult of all from a statistical viewpoint and that more accurate signal detection for a given level of signal sparsity and strength can be obtained when correlation is present we characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary the boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is toeplitz

optimal switching for ordinary differential equations we consider the problem of controlling an ordinary differential equation subject to positive switching costs and show in particular that the value functions form the viscosity solution cf 6 7 of the dynamic programming quasi variational inequalities this interpretation allows for a rigorous application of various dynamic programming techniques

new goodness of fit tests and their application to nonparametric confidence sets suppose one observes a process v on the unit interval where dv f 0 dw with an unknown parameter f 0 l 1 0 1 and standard brownian motion w we propose a particular test of one point hypotheses about f 0 which is based on suitably standardized increments of v this test is shown to have desirable consistency properties if for instance f 0 is restricted to various holder classes of functions the test is mimicked in the context of nonparametric density estimation nonparametric regression and interval censored data under shape restrictions on the parameter such as monotonicity or convexity we obtain confidence sets for f 0 adapting to its unknown smoothness

bootstrapping empirical functions in our previous sections we have established weak convergence of mean residual life total time of test lorenz and goldie processes to appropriate gaussian processes apart from a few special cases cf section 8 these limiting processes are functions of the underlying distributions consequently when testing for statistical hypotheses for example one would have to compute the resulting limiting distribution for each f of interest the same is true when trying to use our results for constructing confidence bands for the theoretical functionals of the said empirical processes this type of problems can be solved by adapting the bootstrap method proposed by efron 1979 1982 to the present situation as we will see the bootstrap method is simply a monte carlo simulation determined by the given observations

estimation of copula based semiparametric time series models this paper studies the estimation of a class of copula based semiparametric stationary markov models these models are characterized by nonparametric marginal distributions and parametric copula functions while the copulas capture all the scale free temporal dependence of the processes simple estimators of the marginal distribution and the copula parameter are provided and their asymptotic properties are established under easily verifiable conditions these results are used to obtain root n consistent and asymptotically normal estimators of important features of the transition distribution such as the nonlinear conditional moments and conditional quantiles the semiparametric conditional quantile estimators are automatically monotonic across quantiles which is attractive for portfolio conditional value at risk calculations

copula based time series with filtered nonstationarity abstract economic and financial time series data can exhibit nonstationary and nonlinear patterns simultaneously this paper studies copula based time series models that capture both patterns we introduce a procedure where nonstationarity is removed via a filtration and then the nonlinear temporal dependence in the filtered data is captured via a flexible markov copula we propose two estimators of the copula dependence parameters the parametric two step copula estimator where the marginal distribution of the filtered series is estimated parametrically and the semiparametric two step copula estimator where the marginal distribution is estimated via a rescaled empirical distribution of the filtered series we show that the limiting distribution of the parametric copula estimator depends on the nonstationary filtration and the parametric marginal distribution estimation and may be non normal surprisingly the limiting distribution of the semiparametric copula estimator using the filtered data is shown to be the same as that without nonstationary filtration which is normal and free of marginal distribution specification the simple and robust properties of the semiparametric copula estimators extend to models with misspecified copulas and facilitate statistical inferences such as hypothesis testing and model selection tests on semiparametric copula based dynamic models in the presence of nonstationarity monte carlo studies and real data applications are presented

sharp rates of convergence of maximum likelihood estimators in nonparametric models sharp rates of convergence of maximum likelihood estimators are established in models which are defined by probability densities having bounded derivatives this result is achieved by making use of local properties of the empirical distribution function

a data driven procedure for density estimation with some applications this paper deals with the probability density estimation using a kernel based approach where the window size of the kernel is found by a data driven procedure it is theoretically shown that under certain assumptions the estimated densities on bounded sets can be asymptotically unbiased when the width of window is obtained from the minimal spanning tree of the observed data the theoretical development initially carried out on r2 is applicable to higher dimensional spaces the results are experimentally verified on bounded sets with different types of distributions the behaviour of the estimator in the case of the unbounded set as in that for gaussian density is also experimentally seen to be good some applications of the proposed density estimation technique is demonstrated one application is the representative point detection algorithm which can be applied for data reduction and outlier rejection another application involves detection of border points of a dot pattern as well as finding a thinned version of the dot pattern

uniqueness of unbounded viscosity solutions for impulse control problem we study here the impulse control problem in infinite as well as finite horizon we allow the cost functionals and dynamics to be unbounded and hence the value function can possibly be unbounded we prove that the value function is the unique viscosity solution in a suitable subclass of continuous functions of the associated quasivariational inequality our uniqueness proof for the infinite horizon problem uses stopping time problem and for the finite horizon problem comparison method however we assume proper growth conditions on the cost functionals and the dynamics

the asymptotic distribution of sums of extreme values from a regularly varying distribution on donne une description complete de la distribution asymptotique de la somme des k n valeurs extremes superieures x n 1 kd n n x n 2 kd n n x n n

comparison theorems for viscosity solutions of a system of quasivariational inequalities with application to optimal control with switching costs in this paper we prove a comparison principle between a viscosity sub and supersolution for a system of quasivariational inequalities and apply it to show that a continuous lower value vector function of an optimal switching cost control problem is characterized as the minimal nonnegative continuous viscosity supersolution of the sqvi

on the convergence rate in the central limit theorem for associated processes on donne des taux de convergence uniforme dans le theoreme central limite pour des processus associes au troisieme moment fini la stationarite n etant pas requise

connectivity in one dimensional soft random geometric graphs in this paper we study the connectivity of a one dimensional soft random geometric graph rgg the graph is generated by placing points at random on a bounded line segment and connecting pairs of points with a probability that depends on the distance between them we derive bounds on the probability that the graph is fully connected by analyzing key modes of disconnection in particular analytic expressions are given for the mean and variance of the number of isolated nodes and a sharp threshold established for their occurrence bounds are also derived for uncrossed gaps and it is shown analytically that uncrossed gaps have negligible probability in the scaling at which isolated nodes appear this is in stark contrast to the hard rgg in which uncrossed gaps are the most important factor when considering network connectivity

the asymptotic distribution of trimmed sums on donne une condition necessaire et suffisante pour l existence de constantes de normalisation et de centrage

limit theorems and inequalities for the uniform empirical process indexed by intervals the uniform empirical process u is considered as a process indexed by intervals powerful new exponential bounds are established for the process indexed by both points and intervals these bounds trivialize the proof of the chibisov o reilly theorem concerning the convergence of the process with respect to 11 q 11 metrics and are used to prove an interval analogue of the chibisov o reilly theorem a strong limit theorem related to the well known holder condition for brownian bridge u is also proved connections with related work of csdki eicker jaeschke and stute are mentioned as an application we introduce a new statistic for testing uniformity which is the natural interval analogue of the classical anderson darling statistic

on the erdos renyi theorem for random fields and sequences and its relationships with the theory of runs and spacings we prove in this paper a law of erdos renyi type for arrays of independent and identically distributed random variables the relationships of our theorem with similar results obtained in the theory of runs and spacings are investigated applications include the evaluation of the rate of convergence of erdos renyi maxima in limiting cases and a generalization of a theorem of erdos and revesz on runs

strong laws for the k th order statistic when k c log2n we give several bounds and characterizations for the lower almost sure classes of uk n where k kn denotes a nondecreasing integer sequence such that 1 kn 0 log2n as n and where u1 n un n denote the order statistics of the first n observations from a sequence of independent and uniformly distributed random variables on 0 1

strong bounds for multidimensional spacings let u1 u2 u n be independent random vectors uniformly distributed on 0 1 d we define the k th maximal spacing n k associated to u1 u1 as the k th maximal possible value of the length of a side of a d dimensional square block in 0 1 d which does not intersect the sample and which cannot be enlarged without doing so our main result is that n k n 1 log n 0 log2n 1 d almost surely as n other bounds are proposed for the limiting almost sure behavior of n k as n

the asymptotic distribution of extreme sums converges in distribution along subsequences of the integers n to nondegenerate limits and completely describe the possible subsequential limiting distributions we also give a necessary and sufficient condition for the existence of an and cn such that en be asymptotically normal along a given subsequence and with suitable an and cn determine the limiting distributions of en along the whole sequence n when f is in the domain of attraction of an extreme value distribution

semiparametric trending panel data models with cross sectional dependence abstract a semiparametric fixed effects model is introduced to describe the nonlinear trending phenomenon in panel data analysis and it allows for the cross sectional dependence in both the regressors and the residuals a pooled semiparametric profile likelihood dummy variable approach based on the first stage local linear fitting is developed to estimate both the parameter vector and the nonlinear time trend function as both the time series length t and the cross sectional size n tend to infinity the resulting estimator of the parameter vector is asymptotically normal with a root n t convergence rate meanwhile the asymptotic distribution for the nonparametric estimator of the trend function is also established with a root n t h convergence rate two simulated examples are provided to illustrate the finite sample performance of the proposed method in addition the proposed model and estimation method are applied to a cpi data set as well as an input output data set

weak convergence for weighted empirical processes of dependent sequences in this paper we establish weak convergence theorems for weighted empirical processes of strong mixing p mixing and associated sequences we apply these results to obtain weak convergence of integral functionals of empirical processes and of mean residual life processes in reliability theory to carry out the proofs we develop two rosenthal type inequalities for strong mixing and associated sequences

the dip test of unimodality the dip test measures multimodality in a sample by the maximum difference over all sample points between the empirical distribution function and the unimodal distribution function that minimizes that maximum difference the uniform distribution is the asymptotically least favorable unimodal distribution and the distribution of the test statistic is determined asymptotically and empirically when sampling from the uniform

one dimensional geometric random graphs with nonvanishing densities part i a strong zero one law for connectivity we consider a collection of n independent points which are distributed on the unit interval 0 1 according to some probability distribution function f two nodes are said to be adjacent if their distance is less than some given threshold value when f admits a nonvanishing density f we show under a weak continuity assumption on f that the property of graph connectivity for the induced geometric random graph exhibits a strong zero one law and we identify the corresponding critical scaling this is achieved by generalizing to nonuniform distributions a limit result obtained by levy for maximal spacings under the uniform distribution

strong invariance principles for sequential bahadur kiefer and vervaat error processes of long range dependent sequences in this paper we study strong approximations invariance principles of the sequential uniform and general bahadur kiefer processes of long range dependent sequences we also investigate the strong and weak asymptotic behavior of the sequential vervaat process that is the integrated sequential bahadur kiefer process properly normalized as well as that of its deviation from its limiting process the so called vervaat error process it is well known that the bahadur kiefer and the vervaat error processes cannot converge weakly in the i i d case in contrast to this we conclude that the bahadur kiefer and vervaat error processes as well as their sequential versions do converge weakly to a dehling taqqu type limit process for certain long range dependent sequences

strong laws for the maximal k spacing when k c log n we consider the maximal k spacing m n mathop max limits 0 underset raise0 3em hbox smash scriptscriptstyle leqslant i underset raise0 3em hbox smash scriptscriptstyle leqslant n 1 k left u ni k u ni right where u n1 u nn are the order statistics of an i i d sample of size n from the uniform distribution on 0 1 and u n0 0 u nn 1 1 the integer k is allowed to vary with n at a rate not exceeding log n we obtain laws of the iterated logarithm for all the k s in the given range for small k the methods used in the proofs are borrowed from extreme value theory for larger k the techniques are reminiscent of those used in the proof of the erdos renyi theorem

cramer type moderate deviations for self normalized ψ mixing sequences abstract let i i 1 be a sequence of mixing random variables let m n 0 1 k n 2 m and y j i 1 m m j 1 i 1 j k set s k o j 1 k y j and s o k i 1 k y j 2 we prove a cramer type moderate deviation expansion for p s k o s o k x as n our result is similar to the recent work of chen et al 2016 4 where the authors established cramer type moderate deviation expansions for mixing sequences comparing to the result of chen et al our results hold for mixing coefficients with polynomial decaying rate and wider ranges of validity

kernel type estimators for the extreme value index a large part of the theory of extreme value index estimation is developed for positive extreme value indices the best known estimator of a positive extreme value index is probably the hill estimator this estimator belongs to the category of moment estimators but can also be interpreted as a quasi maximum likelihood estimator it has been generalized to a kernel type estimator but this kernel type estimator can similarly to the hill estimator only be used for the estimation of positive extreme value indices in the present paper we introduce kernel type estimators which can be used for estimating the extreme value index over the whole positive and negative range we present a number of results on their distributional behavior and compare their performance with the performance of other estimators such as moment type estimators for the whole range and the quasi maximum likelihood estimator based on the generalized pareto distribution we also discuss an automatic bandwidth selection method and introduce a kernel estimator for a second order parameter controlling the speed of convergence

the controversial stability analysis in this review we present different techniques for obtaining stability limits for a finite difference scheme the forward time and space centered numerical scheme applied to the convection diffusion equation a survey of past attempts to state stability conditions for this scheme illustrates the difficulties in stability analysis that arise as soon as a scheme becomes more complex and illuminates the concepts of necessary and sufficient conditions for stability

embedding the finite sampling process at a rate a huge body of if and only if theorems can be obtained based on certain strong embedding theorems for the partial sum process mathbb s n and the uniform empirical and quantile processes mathbb u n and mathbb v n this embedding was accomplished in 1986 by m csorgo s csorgo l horvath and d mason their embedding is beautifully formulated so that many necessary and sufficient type results can be established using it it is worthwhile to have an accessible proof indeed these authors have since produced two papers that obtain the essential form of their result without appealing to earlier work of j komlos p major and g tusnady indeed this present paper does this for the finite sampling process mathbb r n and the weighted empirical process mathbb w n these latter results are entirely new and are the main objectives of the present paper the applications of these latter results will appear elsewhere

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

reduction principles for quantile and bahadur kiefer processes of long range dependent linear sequences in this paper we consider quantile and bahadur kiefer processes for long range dependent linear sequences these processes unlike in previous studies are considered on the whole interval 0 1 as it is well known quantile processes can have very erratic behavior on the tails we overcome this problem by considering these processes with appropriate weight functions in this way we conclude strong approximations that yield some remarkable phenomena that are not shared with i i d sequences including weak convergence of the bahadur kiefer processes a different pointwise behavior of the general and uniform bahadur kiefer processes and a somewhat strange behavior of the general quantile process

upper bounds for k th maximal spacings if kn denotes the k th maximal spacing generated by an i i d sequence of random variables uniformly distributed on 0 1 we show that for any p 3 p left nk n underset raise0 3em hbox smash scriptscriptstyle geqslant text log n left 2 text log text 2 n text log text 3 n ldots left 1 varepsilon right text log p n right k text i text o text right 1 text or 0 according as e 0 or e 0 we also obtain strong limiting bounds for the record times and inter record times of kn

on some simple estimates of an exponent of regular variation summary we present a class of simple estimates of an exponent of regular variation unlike those proposed recently by de haan and resnick 1980 ours converge at an algebraic rather than a logarithmic rate and unlike those considered by teugels 1980 their form does not depend on the unknown exponent

a refinement of the kmt inequality for the uniform empirical process a refinement of the komlos major and tusnady 1975 inequality for the supremum distance between the uniform empirical process and a constructed sequence of brownian bridges is obtained this inequality leads to a weighted approximation of the uniform empirical and quantile processes by a sequence of brownian bridges dual to that recently given by m csoorg s csorg horvath and mason 1986 the present theory approximates the uniform empirical process more closely than the uniform quantile process whereas the former theory more closely approximates the uniform quantile process

on the strong convergence rate for positively associated random variables abstract for sequences of the positively associated random variables which are strictly weaker than the classical associated random ones introduced by esary et al 1967 7 strong convergence rate n 1 2 log n 1 2 is obtained which reaches the available one for independent random variables in terms of berstein type inequality further we give the corresponding precise asymptotics with respect to the rate mentioned above which extend and improve the relevant results in fu 2009 8

adaptive inference for a semiparametric generalized autoregressive conditional heteroskedasticity model abstract this paper considers a semiparametric generalized autoregressive conditional heteroskedasticity s garch model for this model we first estimate the time varying long run component for unconditional variance by the kernel estimator and then estimate the non time varying parameters in garch type short run component by the quasi maximum likelihood estimator qmle we show that the qmle is asymptotically normal with the parametric convergence rate next we construct a lagrange multiplier test for linear parameter constraint and a portmanteau test for model checking and obtain their asymptotic null distributions our entire statistical inference procedure works for the non stationary data with two important features first our qmle and two tests are adaptive to the unknown form of the long run component second our qmle and two tests share the same efficiency and testing power as those in variance targeting method when the s garch model is stationary

central limit theorems for lp norms of density estimators let x1 x2 be a sequence of independent identically distributed random variables with a smooth density function f we obtain central limit theorems for int limits infty infty f n t f t p d mu t text 1 underset raise0 3em hbox smash scriptscriptstyle leqslant p infty where fn is a kernel estimate of f and is a measure on the borel sets of

strong limit theorems for oscillation moduli of the uniform empirical process let u n t n tfrac 1 2 gamma n t t 0 leqq t leqq 1 denote the uniform empirical process based on the first n of a sequence 1 2 of iid uniform 0 1 random variables where gamma n t n 1 sum limits i 1 n 1 0 t xi i is the empirical distribution function the oscillation modulus of u n is defined by omega n a sup u n t h u n t 0 leqq t leqq 1 h h leqq a and the lipschitz 1 2 modulus of u n is defined by tilde omega n a sup u n t h u n t h tfrac 1 2 0 leqq t leqq 1 h a leqq h leqq 1

distribution on warp maps for alignment of open and closed curves alignment of curve data is an integral part of their statistical analysis and can be achieved using model or optimization based approaches the parameter space is usually the set of monotone con

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

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

quasi variational inequalities and first order hamilton jacobi equations on considere des inegalites quasi variationnelles de hamilton jacobi dans r n on introduit pour ce probleme la notion de solution viscosite

stability analysis of finite difference schemes for the advection diffusion equation we present a collection of stability results for finite difference approximations to the advection diffusion equation u t a u x b u xx the results are for centered difference schemes in space and include explicit and implicit schemes in time up to fourth order and schemes that use different space and time discretizations for the advective and diffusive terms the results are derived from a uniform framework based on the schur cohn theory of simple von neumann polynomials and are necessary and sufficient for the stability of the cauchy problem some of the results are believed to be new

inference for the limiting cluster size distribution of extreme values any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily compound poisson under appropriate long range dependence conditions typically exceedances appear in clusters the underlying poisson points represent the cluster positions and the multiplicities correspond to the cluster sizes in the present paper we introduce estimators of the limiting cluster size probabilities which are constructed through a recursive algorithm we derive estimators of the extremal index which plays a key role in determining the intensity of cluster positions we study the asymptotic properties of the estimators and investigate their finite sample behavior on simulated data

conditional versions of limit theorems for conditionally associated random variables abstract relation between association and conditional association is answered several examples show that the association of random variables does not imply the conditional association and vice versa several fundamental properties of conditional associated random variables are developed which extend the corresponding ones under the non conditioning setup by means of these properties some conditional hajek renyi type inequalities a conditional strong law of large numbers and a conditional central limit theorem stated in terms of conditional characteristic functions are established which are conditional versions of the earlier results for associated random variables respectively in addition some lemmas in the context are of independent interest

a two stage hybrid procedure for estimating an inverse regression function we consider a two stage procedure tsp for estimating an inverse regression function at a given point where isotonic regression is used at stage one to obtain an initial estimate and a local linear approximation in the vicinity of this estimate is used at stage two we establish that the convergence rate of the second stage estimate can attain the parametric n 1 2 rate furthermore a bootstrapped variant of tsp btsp is introduced and its consistency properties studied this variant manages to overcome the slow speed of the convergence in distribution and the estimation of the derivative of the regression function at the unknown target quantity finally the finite sample performance of btsp is studied through simulations and the method is illustrated on a data set

hammerstein system identification with the nearest neighbor algorithm the nonlinear characteristic in a hammerstein system i e a system in which a nonlinear memoryless subsystem and a linear dynamic are connected in a cascade is recovered with the nonparametric nearest neighbor regression estimate the a priori information is nonparametric both the nonlinear characteristic and the impulse response are completely unknown and can be of any form local and global properties of the estimate are examined whatever the probability density of the input signal the estimate converges at every continuity point of the characteristic as well as in the global sense we derive the asymptotic bias and variance of the proposed estimate as a result the optimal rate of convergence is established that additionally is independent of the shape of the input density results of numerical simulations are also presented

optimal consumption until ruin for an endowment described by an autonomous ode for an infinite time horizon we give an algorithmic solution of the optimal consumption problem supc 0 e tdct where ct denotes the accumulated consumption until time t and denotes the time of ruin moreover the endowment process xt is modeled by xt x 0t xs ds ct we solve the problem by showing that the function provided by the algorithm solves the hamilton jacobi hj equation in a viscosity sense and that the same is true for the value function of the problem the argument is finished by a uniqueness result it turns out that one has to change the optimal strategy at a sequence of endowment values described by a free boundary value problem finally we give an illustrative example

estimating covariation epps effect microstructure noise abstract this paper is about how to estimate the integrated covariance x y t of two assets over a fixed time horizon 0 t when the observations of x and y are contaminated and when such noisy observations are at discrete but not synchronized times we show that the usual previous tick covariance estimator is biased and the size of the bias is more pronounced for less liquid assets this is an analytic characterization of the epps effect we also provide the optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms including nonsynchronous trading microstructure noise and time discretization finally a two scales covariance estimator is provided which simultaneously cancels to first order the epps effect and the effect of microstructure noise the gain is demonstrated in data

laws of the iterated logarithm for order statistics of uniform spacings let x 1 x 2 cdots be a sequence of independent uniformly distributed random variables on lbrack 0 1 rbrack and let k n be the k th largest spacing induced by the order statistics of x 1 cdots x n 1 we show that lim sup nk n log n 2 log 2n 1 k quad text almost surely and lim inf nk n log n log 3n c quad text almost surely where log 2 leq c leq 0 and log j is the j times iterated logarithm

self normalized cramer type moderate deviations under dependence we establish a cramer type moderate deviation result for self normalized sums of weakly dependent random variables where the moment requirement is much weaker than the non self normalized counterpart the range of the moderate deviation is shown to depend on the moment condition and the degree of dependence of the underlying processes we consider three types of self normalization the equal block scheme the big block small block scheme and the interlacing scheme simulation study shows that the latter can have a better finite sample performance our result is applied to multiple testing and construction of simultaneous confidence intervals for ultra high dimensional time series mean vectors

functional coefficient regression models for nonlinear time series abstract the local linear regression technique is applied to estimation of functional coefficient regression models for time series data the models include threshold autoregressive models and functional coefficient autoregressive models as special cases but with the added advantages such as depicting finer structure of the underlying dynamics and better postsample forecasting performance also proposed are a new bootstrap test for the goodness of fit of models and a bandwidth selector based on newly defined cross validatory estimation for the expected forecasting errors the proposed methodology is data analytic and of sufficient flexibility to analyze complex and multivariate nonlinear structures without suffering from the curse of dimensionality the asymptotic properties of the proposed estimators are investigated under the mixing condition both simulated and real data examples are used for illustration

smooth fit principle for impulse control of multidimensional diffusion processes value functions of impulse control problems are known to satisfy quasi variational inequalities qvis a bensoussan and j l lions impulse control and quasivariational inequalities heyden son philadelphia 1984 translation of controle impulsionnel et inequations quasi variationnelles gauthier villars paris 1982 this paper proves the smooth fit c 1 property of the value function for multidimensional controlled diffusions using a viscosity solution approach we show by examples how to exploit this regularity property to derive explicitly optimal policy and value functions

deterministic impulse control problems we prove that the optimal cost function of a deterministic impulse control problem is the unique viscosity solution of a first order hamilton jacobi quasi variational inequality in mathbb r n