inference in dynamic models containing surprise variables abstract some new results on calculating moving average representation mar coefficients and their limiting distribution from estimated vector arma processes are presented the technique is applied to the problem of estimating the coefficients of unanticipated or surprise variables in a single equation for a multi period expectations horizon the method naturally conditions the expectations on all past values of the process and avoids the necessity of using two step regression procedures and adjusting the resulting standard errors

second order asymptotics for the gaussian mac with degraded message sets this paper studies the second order asymptotics of the gaussian multiple access channel with degraded message sets for a fixed average error probability varepsilon in 0 1 and an arbitrary point on the boundary of the capacity region we characterize the speed of convergence of rate pairs that converge to that boundary point for codes that have asymptotic error probability no larger than varepsilon as a stepping stone to this local notion of the second order asymptotics we study a global notion and establish relationships between the two we provide a numerical example to illustrate how the angle of approach to a boundary point affects the second order coding rate this is the first conclusive characterization of the second order asymptotics of a network information theory problem in which the capacity region is not a polygon

common message broadcast channels with feedback in the nonasymptotic regime full feedback we investigate the maximum coding rate for a given average blocklength and error probability over a k user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable length stop feedback codes for the point to point case polyanskiy et al 2011 demonstrated that variable length coding combined with stop feedback significantly increases the speed of convergence of the maximum coding rate to capacity this speed up manifests itself in the absence of a square root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths i e zero dispersion in this paper we present nonasymptotic achievability and converse bounds on the maximum coding rate of the common message k user discrete memoryless broadcast channel which strengthen and generalize the ones reported in trillingsgaard et al 2015 for the two user case an asymptotic analysis of these bounds reveals that zero dispersion cannot be achieved for certain common message broadcast channels e g the binary symmetric broadcast channel furthermore we identify conditions under which our converse and achievability bounds are tight up to the second order through numerical evaluations we illustrate that our second order expansions approximate accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point to point case

strong noise sensitivity and random graphs the noise sensitivity of a boolean function describes its likelihood to flip under small perturbations of its input introduced in the seminal work of benjamini kalai and schramm 1999 it was there shown to be governed by the first level of fourier coefficients in the central case of monotone functions at a constant critical probability here we study noise sensitivity and a natural stronger version of it addressing the effect of noise given a specific witness in the original input our main context is the erdos renyi random graph where already the property of containing a given graph is sufficiently rich to separate these notions in particular our analysis implies strong noise sensitivity in settings where the bks criterion involving the first fourier level does not apply e g when the critical value goes to 0 polynomially fast in the number of variables

initial conditions and moment restrictions in dynamic panel data models estimation of the dynamic error components model is considered using two alternative linear estimators that are designed to improve the properties of the standard first differenced gmm estimator both estimators require restrictions on the initial conditions process asymptotic efficiency comparisons and monte carlo simulations for the simple ar 1 model demonstrate the dramatic improvement in performance of the proposed estimators compared to the usual first differenced gmm estimator and compared to non linear gmm the importance of these results is illustrated in an application to the estimation of a labour demand model using company panel data

the effects of dynamic feedbacks on ls and mm estimator accuracy in panel data models the finite sample behavior is analyzed of particular least squares ls and a range of generalized method of moments mm estimators in panel data models with individual effects and both a lagged dependent variable regressor and another explanatory variable the latter may be affected by lagged feedbacks from the dependent variable too asymptotic expansions indicate how the order of magnitude of bias of mm estimators tends to increase with the number of moment conditions exploited they also provide analytic evidence on how the bias of the various estimators depends on the feedbacks and on other model characteristics such as prominence of individual effects and correlation between observed and unobserved heterogeneity simulation results corroborate the theoretical findings and reveal that in small samples of models with dynamic feedbacks none of the techniques examined dominates regarding bias and mean squared error over all parametrizations examined

approximate p values for local sequence alignments assume that two sequences from a finite alphabet are optimally aligned according to a scoring system that rewards similarities according to a general scoring scheme and penalizes gaps insertions and deletions under the assumption that the letters in each sequence are independent and identically distributed and the two sequences are also independent approximate p values are obtained for the optimal local alignment when either i there are at most a fixed number of gaps or ii the gap initiation cost is sufficiently large in the latter case the approximation can be written in the same form as the well known case of ungapped alignments

hub discovery in partial correlation graphs one of the most important problems in large scale inference problems is the identification of variables that are highly dependent on several other variables when dependence is measured by partial correlations these variables identify those rows of the partial correlation matrix that have several entries with large magnitudes i e hubs in the associated partial correlation graph this paper develops theory and algorithms for discovering such hubs from a few observations of these variables we introduce a hub screening framework in which the user specifies both a minimum partial correlation rho and a minimum degree delta to screen the vertices the choice of rho and delta can be guided by our mathematical expressions for the phase transition correlation threshold rho c governing the average number of discoveries they can also be guided by our asymptotic expressions for familywise discovery rates under the assumption of large number p of variables fixed number n of multivariate samples and weak dependence under the null hypothesis that the dispersion covariance matrix is sparse these limiting expressions can be used to enforce familywise error constraints and to rank the discoveries in order of increasing statistical significance for n ll p the computational complexity of the proposed partial correlation screening method is low and is therefore highly scalable thus it can be applied to significantly larger problems than previous approaches the theory is applied to discovering hubs in a high dimensional gene microarray dataset

on latin hypercube sampling this paper contains a collection of results on latin hypercube sampling the first result is a berry esseen type bound for the multivariate central limit theorem of the sample mean hat mu n based on a latin hypercube sample the second establishes sufficient conditions on the convergence rate in the strong law for hat mu n finally motivated by the concept of empirical likelihood a way of constructing nonparametric confidence regions based on latin hypercube samples is proposed for vector means

nonasymptotic and second order achievability bounds for coding with side information we present a novel nonasymptotic or finite blocklength achievability bounds for three side information problems in network information theory these include 1 the wyner ahlswede korner wak problem of almost lossless source coding with rate limited side information 2 the wyner ziv wz problem of lossy source coding with side information at the decoder and 3 the gel fand pinsker gp problem of channel coding with noncausal state information available at the encoder the bounds are proved using ideas from channel simulation and channel resolvability our bounds for all three problems improve on all previous nonasymptotic bounds on the error probability of the wak wz and gp problems in particular those derived by verdu using our novel nonasymptotic bounds we recover the general formulas for the optimal rates of these side information problems finally we also present achievable second order coding rates by applying the multidimensional berry esseen theorem to our new nonasymptotic bounds numerical results show that the second order coding rates obtained using our nonasymptotic achievability bounds are superior to those obtained using existing finite blocklength bounds

an extreme value theory for long head runs for an infinite sequence of independent coin tosses withp heads p 0 1 the longest run of consecutive heads in the firstn tosses is a natural object of study we show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum ofn 1 p i i d exponential random variables these results are extended to the case of the longest head run interrupted byk tails the mean length of this run is shown to be log n klog n k 1 log 1 p log k k 1 2 r1 n o 1 where log log1 p 0 577 is the euler mascheroni constant ln 1 p andr1 n is small the variance is 2 6 2 1 12 r2 n o 1 wherer2 n is again small upper and lower class results for these run lengths are also obtained and extensions discussed

sunklodas approach to normal approximation for time dependent dynamical systems we consider time dependent dynamical systems arising as sequential compositions of self maps of a probability space we establish conditions under which the birkhoff sums for multivariate observations given a centering and a general normalizing sequence b n of invertible square matrices are approximated by a normal distribution with respect to a metric of regular test functions depending on the metric and the normalizing sequence b n the conditions imply that the error in the approximation decays either at the rate o n 1 2 or the rate o n 1 2 log n under the additional assumption that vert b n 1 vert lesssim n 1 2 the error comes with a multiplicative constant whose exact value can be computed directly from the conditions the proof is based on an observation due to sunklodas regarding stein s method of normal approximation we give applications to one dimensional random piecewise expanding maps and to sequential random and quasistatic intermittent systems

extending the scope of empirical likelihood this article extends the scope of empirical likelihood methodology ill three directions to allow for plug in estimates of nuisance parameters in estimating equations slower than root n rates of convergence and settings in which there are a relatively large number of estimating equations compared to the sample size calibrating empirical likelihood confidence regions with plug in is sometimes intractable due to the complexity of the asymptotics so we introduce a bootstrap approximation that call be used in such situations we provide a range of examples from survival analysis and nonparametric statistics to illustrate the main results

multivariate approximations in wasserstein distance by stein s method and bismut s formula stein s method has been widely used for probability approximations however in the multi dimensional setting most of the results are for multivariate normal approximation or for test functions with bounded second or higher order derivatives for a class of multivariate limiting distributions we use bismut s formula in malliavin calculus to control the derivatives of the stein equation solutions by the first derivative of the test function combined with stein s exchangeable pair approach we obtain a general theorem for multivariate approximations with near optimal error bounds on the wasserstein distance we apply the theorem to the unadjusted langevin algorithm

asymptotic distributions of impulse response functions in short panel vector autoregressions this paper establishes the asymptotic distributions of the impulse response functions in panel vector autoregressions with a fixed time dimension it also proves the asymptotic validity of a bootstrap approximation to their sampling distributions the autoregressive parameters are estimated using the gmm estimators based on the first differenced equations and the error variance is estimated using an extended analysis of variance type estimator contrary to the time series setting we find that the gmm estimator of the autoregressive coefficients is not asymptotically independent of the error variance estimator the asymptotic dependence calls for variance correction for the orthogonalized impulse response functions simulation results show that the variance correction improves the coverage accuracy of both the asymptotic confidence band and the studentized bootstrap confidence band for the orthogonalized impulse response functions

poisson and compound poisson approximations in conventional and nonconventional setups it was shown in kifer israel j math 2013 that for any subshift of finite type considered with a gibbs invariant measure the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of almost all points are asymptotically poisson distributed here we not only extend this result to all psi mixing shifts with countable alphabet but actually show that for all points the distributions of these numbers are asymptotically close either to poisson or to compound poisson distributions it turns out that for all nonperiodic points a limiting distribution is always poisson while at the same time for periodic points there may be no limiting distribution at all unless the shift invariant measure is bernoulli in which case the limiting distribution always exists thus we describe essentially completely limiting distributions of multiple recurrence numbers in this setup as a corollary we obtain also that the first occurence time of the multiple recurrence event is asymptotically exponentially distributed most of the results are new also for the widely studied single recurrencies case see for instance haydn and vaienti discret contin dyn syst a 10 589 616 2004 probab theory relat fields 144 517 542 2009 abadi and saussol stoch process appl 121 314 323 2011 abadi and vergne nonlinearity 21 2871 2885 2008 as well

a note on the asymptotic distribution of impulse response functions of estimated var models with orthogonal residuals abstract a simple closed form expression is given for the covariance matrix of the asymptotic distribution of the coefficients of a moving average representation with orthogonal residuals computed from an estimated vector autoregressive process

critical phenomena in sequence matching we give a generalization of the result of erdos and renyi on the length r n of the longest head run in the first n tosses of a coin consider two independent sequences x 1 x 2 cdots x m and y 1y 2 cdots y n suppose that x 1 x 2 cdots are i i d mu and y 1 y 2 cdots are i i d nu where mu and nu are possibly different distributions on a common finite alphabet s let p equiv p x 1 y 1 in 0 1 the length of the longest matching consecutive subsequence is m m n equiv max k x i r y j r for r 1 to k for some 0 leq i leq m k 0 leq j leq n k for m and n rightarrow infty with log m log mn rightarrow lambda in 0 1 our result is that there is a constant k equiv k mu nu lambda in 0 1 rbrack such that p lim m m n log 1 p mn k 1 the proof uses large deviation methods the constant k is determined from a variational formula involving the kullback liebler distance or relative entropy a simple necessary and sufficient condition for k 1 is given for the case m n lambda 1 2 and mu nu k 1 the set of mu nu lambda for which k 1 has nonempty interior the boundary of this set is the location of a phase transition the results generalize to more than two sequences and to markov chains a strong law of large numbers is given for the proportion of letters within the longest matching word the limiting proportion exhibits critical behavior similar to that of k

stein s method and a quantitative lindeberg clt for the fourier transforms of random vectors abstract we use a multivariate version of stein s method to establish a quantitative lindeberg clt for the fourier transforms of random n vectors we achieve this conceptually mainly by constructing a natural approach structure on n random vectors overlying the topology of weak convergence and technically mainly by deducing a specific integral representation for the hessian matrix of a solution to the stein equation with test function e t x exp i k 1 n t k x k where t x r n

first passage times for threshold growth dynamics on bf z sp 2 in the threshold growth model on an integer lattice the occupied set grows according to a simple local rule a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood in this paper we analyze the behavior of two dimensional threshold growth dynamics started from a sparse bernoulli density of occupied sites we explain how nucleation of rare centers invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case we also briefly address scaling laws for the critical case

spin glasses and stein s method we introduce some applications of stein s method in the high temperature analysis of spin glasses stein s method allows the direct analysis of the gibbs measure without having to eate a cavity another advantage is that it gives limit theorems with total variation error bounds although the bounds can be suboptimal a surprising byproduct of our analysis is a relatively transparent explanation of the thouless anderson palmer system of equations along the way we develop stein s method for mixtures of two gaussian densities

a berry esseen bound for symmetric statistics the rate of convergence of the distribution function of a symmetric function of n independent and identically distributed random variables to its normal limit is investigated under appropriate moment conditions the rate is shown to be o left n frac 1 2 right this theorem generalizes many known results for special cases and two examples are given possible further extensions are indicated

orthogonal polynomials in stein s method stein s method provides a way of finding approximations to the distribution say of a random variable which at the same time gives estimates of the approximation error involved in essence the method is based on a defining equation or equivalently an operator of the distribution and a related stein equation up to now it was not clear which equation to take one could think of a lot of equations we show how for a broad class of distributions there is one convenient equation we give a systematic treatment including the stein equation and its solution a key tool in stein s theory is the generator method developed by barbour and we suggest employing the generator of a markov process for the operator of the stein equation for a given distribution there may be various markov processes which fit in barbour s method and up to now it was still not clear which markov process to take to obtain good results we show how for a broad class of distributions there is a special markov process a birth and death process or a diffusion which takes a leading role in the analysis furthermore a key role is played by the classical orthogonal polynomials it turns out that the defining operator is based on a hypergeometric difference or differential equation which lies at the heart of the classical orthogonal polynomials furthermore the spectral representation of the transition probabilities of the markov process involved are in terms of orthogonal polynomials closely related to the distribution to be approximated this systematic treatment together with the introduction of orthogonal polynomials in the analysis seems to be new furthermore some earlier uncovered examples like the beta the student s t and the hypergeometric distribution are now worked out

on success runs of a fixed length in bernoulli sequences exact and asymptotic results consider a sequence of n bernoulli success failure or 1 0 trials the exact and limiting distribution of the random variable e n k denoting the number of success runs of a fixed length k email protected email protected n is derived along with its mean and variance an associated waiting time is examined as well the exact distribution is given in terms of binomial coefficients and an extension of it covering exchangeable sequences is also discussed limiting distributions of e n k are obtained using poisson and normal approximations the exact mean and variance of e n k which are given in explicit forms are also used to derive bounds and an additional approximation of the distribution of e n k numbers associated with e n k and related random variables counting binary strings and runs of 1 s useful in applications of computer science are provided the overall study is illustrated by an extensive numerical experimentation

on a functional contraction method methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space our approach is an extension of the so called contraction method to the space c 0 1 of continuous functions endowed with uniform topology and the space d 0 1 of cadlag functions with the skorokhod topology the contraction method originated from the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences it is based on stochastic fixed point equations where probability metrics can be used to obtain contraction properties and allow the application of banach s fixed point theorem we develop the use of the zolotarev metrics on the spaces c 0 1 and d 0 1 in this context applications are given in particular a short proof of donsker s functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed

selective inference with a randomized response inspired by sample splitting and the reusable holdout introduced in the field of differential privacy we consider selective inference with a randomized response we discuss two major advantages of using a randomized response for model selection first the selectively valid tests are more powerful after randomized selection second it allows consistent estimation and weak convergence of selective inference procedures under independent sampling we prove a selective or privatized central limit theorem that transfers procedures valid under asymptotic normality without selection to their corresponding selective counterparts this allows selective inference in nonparametric settings finally we propose a framework of inference after combining multiple randomized selection procedures we focus on the classical asymptotic setting leaving the interesting high dimensional asymptotic questions for future work

stein s method for diffusion approximations stein s method of obtaining distributional approximations is developed in the context of functional approximation by the wiener process and other gaussian processes an appropriate analogue of the one dimensional stein equation is derived and the necessary properties of its solutions are established the method is applied to the partial sums of stationary sequences and of dissociated arrays to a process version of the wald wolfowitz theorem and to the empirical distribution function

the topology of probability distributions on manifolds let mathcal p be a set of n random points in mathbb r d generated from a probability measure on a m dimensional manifold mathcal m subset mathbb r d in this paper we study the homology of mathcal u mathcal p r the union of d dimensional balls of radius r around mathcal p as n rightarrow infty and r rightarrow 0 in addition we study the critical points of d mathcal p the distance function from the set mathcal p these two objects are known to be related via morse theory we present limit theorems for the betti numbers of mathcal u mathcal p r as well as for number of critical points of index k for d mathcal p depending on how fast r decays to zero as n grows these two objects exhibit different types of limiting behavior in one particular case n r m ge c log n we show that the betti numbers of mathcal u mathcal p r perfectly recover the betti numbers of the original manifold mathcal m a result which is of significant interest in topological manifold learning

high order approximation of the frobenius perron operator we formulate a general convergence theory for the finite dimensional projection approximation of the fixed point of a class of linear operators in particular we explore the relation with the standard framework of stability and consistency of the approximation imply convergence applications include the frobenius perron operator p s corresponding to a nonsingular measurable transformation from 0 1 to itself numerical experiments are also given to support the theoretical analysis

formulation and estimation of dynamic models using panel data abstract this paper presents a statistical analysis of time series regression models for longitudinal data with and without lagged dependent variables under a variety of assumptions about the initial conditions of the processes being analyzed the analysis demonstrates how the asymptotic properties of estimators of longitudinal models are critically dependent on the manner in which samples become large by expanding the number of observations per person holding the number of people fixed or by expanding the number of persons holding the number of observations per person fixed the paper demonstrates which parameters can and cannot be identified from data produced by different sampling plans

entropy bounds for discrete random variables via maximal coupling this paper derives new bounds on the difference of the entropies of two discrete random variables in terms of the local and total variation distances between their probability mass functions the derivation of the bounds relies on maximal coupling and they apply to discrete random variables which are defined over finite or countably infinite alphabets loosened versions of these bounds are demonstrated to reproduce some previously reported results the use of the new bounds is exemplified for the poisson approximation where bounds on the local and total variation distances follow from stein s method

on steinʼs method for infinite dimensional gaussian approximation in abstract wiener spaces abstract in this paper we generalize stein s method to infinite variate normal approximation that is an infinite dimensional approximation by abstract wiener measures on a real separable banach space we first establish a stein s identity for abstract wiener measures and solve the corresponding stein s equation then we will present a gaussian approximation theorem using exchangeable pairs in an infinite variate context as an application we will derive an explicit error bound of gaussian approximation to the distribution of a sum of independent and identically distributed banach space valued random variables based on a lindeberg levy type limit theorem in addition an analogous of berry esseen type estimate for abstract wiener measures will be obtained

longest runs in a sequence of m dependent random variables let x i i 1 be a sequence of random 0 s and 1 s we define the lengthr n of the longest head run as follows r n max left k leqq n mathop max limits 0 leqq i leqq n k 1 x i 1 cdots x i k 1 1 right the limit law for the distribution ofr n asn was found in 1943 the rate of convergence in that limit theorem remained unknown until recently even for bernoulli independent trials supposing that x i i 1 is a stationary sequence ofm dependent r v s we given an estimate on the rate of convergence in that limit theorem this estimate provides the unimprovable raten 1 lnn for the independent case a result of lim inf type is obtained too

a weak law of large numbers for empirical measures via stein s method let e be a locally compact hausdorff space with countable basis and let x i i n be a family of random elements on e with 1 n i 1 n l x i h n for a measure with 1 conditions are derived under which l 1 n i 1 n x i n where x denotes the dirac measure at x the proof being based on stein s method there are generalisations that allow for weak dependence between the x i s as examples a dissociated family and an immigration death process are considered the latter illustrates the possible applications in proving convergence of stochastic processes

non normal limiting distribution for optimal alignment scores of strings in binary alphabets we consider two independent binary i i d random strings x and y of equal length n and the optimal alignments according to a symmetric scoring functions only we decompose the space of scoring functions into five components two of these components add a part to the optimal score which does not depend on the alignment and which is asymptotically normal we show that when we restrict the number of gaps sufficiently and add them only into one sequence then the alignment score can be decomposed into a part which is normal and has order o sqrt n and a part which is on a smaller order and tends to a tracy widom distribution adding gaps only into one sequence is equivalent to aligning a string with its descendants in case of mutations and deletes for testing relatedness of strings the normal part is irrelevant since it does not depend on the alignment hence it can be safely removed from the test statistic

asymptotic expansions in the poisson limit theorem asymptotic expansions for the distributions of sums of independent nonnegative integer random variables in the neighbourhood of the poisson distribution are derived together with explicit estimates of the truncation error expansions are also derived for the expectations of at most polynomially growing functions of such sums applications to the poisson binomial and poisson negative binomial approximations are considered the method used is an adaptation of the stein chen approach 1 introduction in chen 1975 stein s method of obtaining error bounds for normal approximations was introduced in the poisson context and was used to obtain rates of convergence in total variation to the poisson distribution for sums of stationary sequences of 0 1 random variables chen also observed that stein s method is in principle suited to developing asymptotic expansions as well as obtaining error estimates and used it for independent 0 1 summands to derive the second term in such an expansion together with an estimate of the remaining error kerstan 1964 had previously derived a similar result with a sharper error estimate by a quite different technique the stein chen method was refined in barbour and hall 1984 to yield the best known upper bounds for the discrepancy between the poisson distribution and that of a sum of independent 0 1 random variables as well as complementary lower bounds and new estimates for the error remaining after the second term in the asymptotic expansion were also established however although stein s method was potentially applicable also to higher order expansions it proved in practice too cumbersome to use in this paper it is shown how by means of a simple identity the stein chen method can be made to yield a full asymptotic expansion with a minimum of difficulty the error estimates obtained perform well in comparison with other estimates that are available for sums of independent 0 1 random variables they agree with those of barbour and hall when the series is truncated after one term or after two for the more general case of sums of independent nonnegative integer valued random variables the error estimates obtained after two terms in the expansion differ only slightly from those in barbour and hall comparison with kerstan s estimates for integral nonnegative summands of the error after one term of the expansion is more complicated by and large the techniques of this paper give better approximations except when the summands themselves are almost precisely poisson distributed in section 3 expansions are also obtained for the expectations of at most polynomially growing functions of the

on the dispersions of three network information theory problems we analyze the dispersions of distributed lossless source coding the slepian wolf problem the multiple access channel and the asymmetric broadcast channel for the two encoder slepian wolf problem we introduce a quantity known as the entropy dispersion matrix which is analogous to the scalar dispersions that have gained interest recently we prove a global dispersion result that can be expressed in terms of this entropy dispersion matrix and provides intuition on the approximate rate losses at a given blocklength and error probability to gain better intuition about the rate at which the nonasymptotic rate region converges to the slepian wolf boundary we define and characterize two operational dispersions 1 the local dispersion and 2 the weighted sum rate dispersion the former represents the rate of convergence to a point on the slepian wolf boundary whereas the latter represents the fastest rate for which a weighted sum of the two rates converges to its asymptotic fundamental limit interestingly when we approach either of the two corner points the local dispersion is characterized not by a univariate gaussian but a bivariate one as well as a subset of off diagonal elements of the aforementioned entropy dispersion matrix finally we demonstrate the versatility of our achievability proof technique by providing inner bounds for the multiple access channel and the asymmetric broadcast channel in terms of dispersion matrices all our proofs are unified by a so called vector rate redundancy theorem which is proved using the multidimensional berry esseen theorem

a berry esseen bound for functions of independent random variables the rate of convergence in the central limit theorem for functions of independent random variables is studied in a unifying approach the basic result sharpens and extends a theorem of van zwet applications to u l and r statistics are also given improving or extending the results of helmers and van zwet helmers and huskova does and van es and helmers

central limit theorems and bootstrap in high dimensions this paper derives central limit and bootstrap theorems for probabilities that sums of centered high dimensional random vectors hit hyperrectangles and sparsely convex sets specifically we derive gaussian and bootstrap approximations for probabilities p n 1 2 ni 1xi a p n 1 2 i 1nxi a where x1 xnx1 xn are independent random vectors in rprp and aa is a hyperrectangle or more generally a sparsely convex set and show that the approximation error converges to zero even if p pn p pn as n n and p np n in particular pp can be as large as o ecnc o ecnc for some constants c c 0c c 0 the result holds uniformly over all hyperrectangles or more generally sparsely convex sets and does not require any restriction on the correlation structure among coordinates of xixi sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments with hyperrectangles being a special case

the erdos renyi strong law for pattern matching with a given proportion of mismatches 1 informal introduction the erdos bnyi law erdijs and r6nyi 1970 l for coin tossing is a strong law for the behavior of the length of long success rich runs for the length r of the longest run of pure heads in n tosses the result is that with probability l lin rjlog n 1 where p flheads 0 a more general result is quoted as formula 1 in section 2 for rz the longest head rich run in which the fraction of heads is at least a p note that r ri in many practical situations such as manufacturing or roulette observations are taken sequentially and each can be classified as success or failure for these cases it is possible to use the erdijs wnyi law to test the hypothesis that the success probability is p from another point of view the erdijs bnyi law can be used to recognize patterns of unusually long runs of succe88e8 or failures our interest is in the recognition of unusually long patterns or words common to two random sequences the patterns are unknown prior to an examination of the sequences the motivation for this work is the comparison of dna sequences which can be modeled as sequences of i i d or markov distributed letters evolution operates to conserve although imperfectly patterns important to biological function it is a task of biology to discover these patterns and their function in earlier work we generalized the erdiie bnyi law for pure head runs a 1 to exact matching patterns between two sequences in this paper we extend those results to include matchings of quality a between p and 1 the examples below illustrate the natural analogs of rz studied in this paper two words form a quality a matching if they have the same length and the

an edgeworth expansion for symmetric statistics we consider asymptotically normal statistics which are symmetric functions of n i i d random variables for these statistics we prove the validity of an edgeworth expansion with remainder o n 1 under cramer s condition on the linear part of the statistic and moment assumptions for all parts of the statistic by means of a counterexample we show that it is generally not possible to obtain an edgeworth expansion with remainder o n 1 without imposing additional assumptions on the structure of the nonlinear part of the statistic

on the rate of convergence in the multivariate clt berry esseen theorems are proved in the multidimensional central limit theorem without using fourier methods an effective and simple estimate of the error in the clt for sums and convex sets using stein s method and induction is derived furthermore the error in the clt for multivariate functions of independent random elements is estimated extending results of van zwet and friedrich to the multivariate case

stein s method on wiener chaos we combine malliavin calculus with stein s method in order to derive explicit bounds in the gaussian and gamma approximations of random variables in a fixed wiener chaos of a general gaussian process our approach generalizes refines and unifies the central and non central limit theorems for multiple wiener ito integrals recently proved in several papers from 2005 to 2007 by nourdin nualart ortiz latorre peccati and tudor we apply our techniques to prove berry esseen bounds in the breuer major clt for subordinated functionals of fractional brownian motion by using the well known mehler s formula for ornstein uhlenbeck semigroups we also recover a technical result recently proved by chatterjee concerning the gaussian approximation of functionals of finite dimensional gaussian vectors

the berry esseen bound for studentized statistics we derive berry esseen bounds for a class of studentized statistics the results are applied to studentized ustatistics studentized l statistics and studentized functions of the sample mean to give the berry esseen bounds under conditions weaker than those obtained by alternative methods

bootstrap confidence sets under model misspecification a multiplier bootstrap procedure for construction of likelihood based confidence sets is considered for finite samples and a possible model misspecification theoretical results justify the bootstrap consistency for a small or moderate sample size and allow to control the impact of the parameter dimension p the bootstrap approximation works if p n is small the main result about bootstrap consistency continues to apply even if the underlying parametric model is misspecified under the so called small modeling bias condition in the case when the true model deviates significantly from the considered parametric family the bootstrap procedure is still applicable but it becomes a bit conservative the size of the constructed confidence sets is increased by the modeling bias we illustrate the results with numerical examples for misspecified constant and logistic regressions jel classification codes c13 c15

asymptotic distribution of the largest off diagonal entry of correlation matrices suppose that we have observations from a dimensional population we are interested in testing that the variates of the population are independent under the situation where goes to infinity as a test statistic is chosen to be where is the sample correlation coefficient between the th coordinate and the th coordinate of the population under an independent hypothesis we prove that the asymptotic distribution of is an extreme distribution of type by using the chen stein poisson approximation method and the moderate deviations for sample correlation coefficients as a statistically more relevant result a limit distribution for where is spearman s rank correlation coefficient between the th coordinate and the th coordinate of the population is derived

a multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments let f 0 1 d r be an integrable function an objective of many computer experiments is to estimate 0 1 d f x dx by f x dx by evaluating f at a finite number of points in 0 1 d there is a design issue in the choice of these points and a popular choice is via the use of randomized orthogonal arrays this article proves a multivariate central limit theorem for a class of randomized orthogonal array sampling designs owen statist sinica 2 1992a 439 452 as well as for a class of oa based latin hypercubes tang j amer statist assoc 81 1993 1392 1397

optimizing randomized trial designs to distinguish which subpopulations benefit from treatment it is a challenge to evaluate experimental treatments where it is suspected that the treatment effect may only be strong for certain subpopulations such as those having a high initial severity of disease or those having a particular gene variant standard randomized controlled trials can have low power in such situations they also are not optimized to distinguish which subpopulations benefit from a treatment with the goal of overcoming these limitations we consider randomized trial designs in which the criteria for patient enrollment may be changed in a preplanned manner based on interim analyses since such designs allow data dependent changes to the population enrolled care must be taken to ensure strong control of the familywise type i error rate our main contribution is a general method for constructing randomized trial designs that allow changes to the population enrolled based on interim data using a prespecified decision rule for which the asymptotic familywise type i error rate is strongly controlled at a specified level as a demonstration of our method we prove new sharp results for a simple two stage enrichment design we then compare this design to fixed designs focusing on each design s ability to determine the overall and subpopulation specific treatment effects

convergence of numerical time averaging and stationary measures via poisson equations numerical approximation of the long time behavior of a stochastic differential equation sde is considered error estimates for time averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the sde the error analysis is based on using an associated poisson equation for the underlying sde the main advantages of this approach are its simplicity and universality it works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables and for hypoelliptic sdes to simplify the exposition we consider only the case where the state space of the sde is a torus and we study only smooth test functions however we anticipate that the approach can be applied more widely an analogy between our approach and stein s method is indicated some practical implications of the results are discussed

estimation of dynamic models with error components abstract observations on n cross section units at t time points are used to estimate a simple statistical model involving an autoregressive process with an additive term specific to the unit different assumptions about the initial conditions are a initial state fixed b initial state random c the unobserved individual effect independent of the unobserved dynamic process with the initial value fixed and d the unobserved individual effect independent of the unobserved dynamic process with initial value random asymptotic properties of the maximum likelihood and covariance estimators are obtained when t and when n the relationship between the pseudo and conditional maximum likelihood estimators is clarified a simple consistent estimator that is independent of the initial conditions and the way in which t or n is also suggested

sur le theoreme de berry esseen pour les suites faiblement dependantes nous etendons la methode de demonstration du theoreme de berry esseen proposee par bergstrom aux suites de variables aleatoires faiblement dependantes en particulier nous montrons que pour les suites stationnaires de variables aleatoires reelles bornees la vitesse de convergence dans le theoreme limite central en distance de levy est de l ordre den 1 2 des que la suite p p 0 des coefficients de melange uniforme satisfait la condition p 0 p p

an extreme value theory for sequence matching on considere des suites finies x 1 x 2 x m et y 1 y 2 y m ou les lettres x i et y i sont choisies i i d sur un alphabet denombrable avec p p x 1 y 1 0 1 on etudie la distribution de la plus longue sequence contigue de couples entre les x et les y avec au plus k mauvais couplages

an erdos renyi law with shifts abstract motivated by the comparison of dna sequences a generalization is given of the result of erdos and renyi on the length r n of the longest run of heads in the first n tosses of a coin consider two sequences x 1 x 2 x n and y 1 y 2 y n the length of the longest matching consecutive subsequence allowing shifts is m n max m x i k y j k for k 1 to m for some 0 i j n m suppose that all the letters are independent and identically distributed the length of the longest match without shifts has the same distribution as r n the length of the longest head run for a biased coin with p p x i y i described by the erdos renyi law p lim n r n log 1 p n 1 1 for matching with shifts our result is p lim n m n log 1 p n 2 1 loosely speaking allowing shifts doubles the length of the longest match the case of markov chains is also handled

poisson convergence for dissociated statistics summary a poisson limit theorem is derived for the number of large values observed among comparisons of independent but not necessarily identically distributed random variables the comparisons made need not be the same and may depend on the two variables being compared an application to the assessment of large numbers of correlation coefficients is given

invariance principles for homogeneous sums universality of gaussian wiener chaos we compute explicit bounds in the normal and chi square approximations of multilinear homogenous sums of arbitrary order of general centered independent random variables with unit variance in particular we show that chaotic random variables enjoy the following form of universality a the normal and chi square approximations of any homogenous sum can be completely characterized and assessed by first switching to its wiener chaos counterpart and b the simple upper bounds and convergence criteria available on the wiener chaos extend almost verbatim to the class of homogeneous sums

markov finite approximation of frobenius perron operator with doubly stochastic tridiagonal action matrix we extend the set of finite rank approximations of frobenius perron operators from projections to general finite rank operators by indicating an arbitrary action matrix we prove the convergence in the case for which the action matrix is double stochastic and tridiagonal and we find the rate of convergence

efficient estimation of models for dynamic panel data abstract in this paper we consider a dynamic model for panel data we show that under standard assumptions there are more moment conditions than are currently exploited in the literature some of these are linear but others are quadratic so that nonlinear gmm is required we also show that exogenous regressors generate a larger number of relevant moment conditions in a dynamic model than they would in a static model

compound poisson approximation for nonnegative random variables via stein s method the aim of this paper is to extend stein s method to a compound poisson distribution setting the compound poisson distributions of concern here are those of the form pois nu where nu is a finite positive measure on 0 infty a number of results related to these distributions are established these in turn are used in a number of examples to give bounds for the error in the compound poisson approximation to the distribution of a sum of random variables

a new method of normal approximation we introduce a new version of stein s method that reduces a large class of normal approximation problems to variance bounding exercises thus making a connection between central limit theorems and concentration of measure unlike skorokhod embeddings the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily as an application we derive a general clt for functions that are obtained as combinations of many local contributions where the definition of local itself depends on the data several examples are given including the solution to a nearest neighbor clt problem posed by p bickel

a high dimensional clt in mathcal w _2 w 2 distance with near optimal convergence rate let x 1 ldots x n be i i d random vectors in mathbb r d with vert x 1 vert le beta then we show that begin aligned frac 1 sqrt n left x 1 cdots x n right end aligned converges to a gaussian in quadratic transportation also known as kantorovich or wasserstein distance at a rate of o left frac sqrt d beta log n sqrt n right improving a result of valiant and valiant the main feature of our theorem is that the rate of convergence is within log n of optimal for n d rightarrow infty

functional poisson approximation in kantorovich rubinstein distance with applications to u statistics and stochastic geometry a poisson or a binomial process on an abstract state space and a symmetric function f acting on k tuples of its points are considered they induce a point process on the target space of f the main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a poisson process on the target space the technical background are a version of stein s method for poisson process approximation a glauber dynamics representation for the poisson process and the malliavin formalism as applications of the main result error bounds for approximations of u statistics by poisson compound poisson and stable random variables are derived and examples from stochastic geometry are investigated

entropy and the fourth moment phenomenon we develop a new method for bounding the relative entropy of a random vector in terms of its stein factors our approach is based on a novel representation for the score function of smoothly perturbed random variables as well as on the de bruijn s identity of information theory when applied to sequences of functionals of a general gaussian field our results can be combined with the carbery wright inequality in order to yield multidimensional entropic rates of convergence that coincide up to a logarithmic factor with those achievable in smooth distances such as the 1 wasserstein distance in particular our findings settle the open problem of proving a quantitative version of the multidimensional fourth moment theorem for random vectors having chaotic components with explicit rates of convergence in total variation that are independent of the order of the associated wiener chaoses the results proved in the present paper are outside the scope of other existing techniques such as for instance the multidimensional stein s method for normal approximations

stein s method for concentration inequalities we introduce a version of stein s method for proving concentration and moment inequalities in problems with dependence simple illustrative examples from combinatorics physics and mathematical statistics are provided

stability of the bakry emery theorem on rn abstract we establish quantitative stability estimates for the bakry emery bound on logarithmic sobolev and poincare constants of uniformly log concave measures more specifically we show that if a 1 uniformly log concave measure has almost the same logarithmic sobolev or poincare constant as the standard gaussian measure then it almost splits off a gaussian factor our results are dimension free leading to dimension free stability estimates for gaussian concentration of lipschitz functions the proofs are based on stein s method optimal transport and an approximate integration by parts identity relating measures and near extremals in the associated functional inequality

extremes of some gaussian random interfaces in this article we give a general criterion for some dependent gaussian models to belong to maximal domain of attraction of gumbel following an application of the stein chen method studied in arratia et al ann probab 17 1 9 25 1989 we also show the convergence of the associated point process as an application we show the conditions are satisfied by some of the well known supercritical gaussian interface models namely membrane model massive and massless discrete gaussian free field fractional gaussian free field

a case where interference does not affect the channel dispersion in 1975 carleial presented a special case of an interference channel called the very strong interference regime in which the interference does not reduce the capacity of the constituent point to point gaussian channels in this paper we show that in the strictly very strong interference regime the dispersions are similarly unaffected more precisely in this paper we characterize the second order coding rates of the gaussian interference channel in the strictly very strong interference regime in other words we characterize the speed of convergence of rates of optimal block codes toward a boundary point of the rectangular capacity region these second order coding rates are expressed in terms of the average probability of error and variances of appropriately defined information densities which coincide with the dispersion of the single user gaussian channel this allows us to conclude that the dispersions are unaffected by interference in this channel model

couplings gradient estimates and logarithmic sobolev inequalitiy for langevin bridges in this paper we establish quantitative results about the bridges of the langevin dynamics and the associated reciprocal processes they include an equivalence between gradient estimates for bridge semigroups and couplings comparison principles bounds of the distance between bridges of different langevin dynamics and a logarithmic sobolev inequality for bridge measures the existence of an invariant measure for the bridges is also discussed and quantitative bounds for the convergence to the invariant measure are proven all results are based on a seemingly new expression of the drift of a bridge in terms of the reciprocal characteristic which roughly speaking quantifies the mean acceleration of a bridge

markov finite approximation of frobenius perron operator the problem of existence and computation of absolutely continuous invariant measures for nonsingular transformations on measure spaces is one of the main concerns in the modern ergodic theory 3 lasota and yorke have established the existence of the invariant measures for a class of nonsingular measurable transformations s from 0 l to itself 4 later in 6 li and yorke gave a sufficient condition for the uniqueness of this invariant density and thus for the ergodicity of the mapping the classical birkhoff s individual ergodic theorem says that if u is an ergodic invariant probability measure under s then for any measurable set a c 0 11 the time average lim 1 n c ax s x which measures the average time spent in a under iteration of s exists to be i for p almost all x where xa is the characteristic function of a 1 on a and 0 off a this seems to suggest that we may use the cesaro sum of the above form to calculate the invariant measure a simple and important example in 5 however shows that the computer round off error can completely dominate the calculation and make the implementation difficult in order to overcome this difficulty li 5 proposed a rigorous numerical procedure which can be implemented on a computer with negligible round off error problem a mapping s 0 l 0 l is called piecewise c2 if there is a partition 0 a a a 1 of 0 l such that for k 1 r the restriction sk of s on uk a is a c2 function which can be extended to the closed interval ak i ak as a c2 function s need not be continuous at the point ak for a c 0 11 we write s a for x s x e a throughout the paper we assume s is a nonsingular measurable transformation i e s is measurable and for any measurable subset a c 0 l with m a 0 m s a 0 here m denotes the lebesgue measure on 0 11 the operator ps l 0 1 l 0 1 defined by