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Showing papers on "Central limit theorem published in 1997"


Journal ArticleDOI
TL;DR: In this article, the convergence of a generalisation of the quadratic variations of a Gaussian process was studied and a convergent estimator of the local Holder index of the sample paths was proposed.
Abstract: We study the convergence of a generalisation of the quadratic variations of a Gaussian process. We build a convergent estimator of the local Holder index of the sample paths and prove a central limit theorem.

394 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the real and imaginary parts of the random variables Tr(Mk), k > 1, converge to independent normal random variables with mean zero and variance k/2, as the size n of the matrix goes to infinity.
Abstract: If M is a matrix taken randomly with respect to normalized Haar measure on U(n), O(n) or Sp(n), then the real and imaginary parts of the random variables Tr(Mk), k > 1, converge to independent normal random variables with mean zero and variance k/2, as the size n of the matrix goes to infinity. For the unitary group this is a direct consequence of the strong Szeg6 theorem for Toeplitz determinants. We will prove a conjecture of Diaconis

214 citations


Journal ArticleDOI
TL;DR: In this article, the authors provide limit theorems for multivariate, possibly non-Gaussian stationary processes whose spectral density matrices may have singularities not restricted at the origin.
Abstract: This paper provides limit theorems for multivariate, possibly non-Gaussian stationary processes whose spectral density matrices may have singularities not restricted at the origin, applying those limiting results to the asymptotic theory of parameter estimation and testing for statistical models of long-range dependent processes. The central limit theorems are proved based on the assumption that the innovations of the stationary processes satisfy certain mixing conditions for their conditional moments, and the usual assumptions of exact martingale difference or the (transformed) Gaussianity for the innovation process are dispensed with. For the proofs of convergence of the covariances of quadratic forms, the concept of the multiple Fejer kernel is introduced. For the derivation of the asymptotic properties of the quasi-likelihood estimate and the quasi-likelihood ratio, the bracketing function approach is used instead of conventional regularity conditions on the model spectral density.

176 citations


Journal ArticleDOI
TL;DR: In this article, a central limit theorem is given for certain weighted partial sums of a covariance stationary process, assuming it is linear in martingale differences, but without any restriction on its spectrum.
Abstract: A central limit theorem is given for certain weighted partial sums of a covariance stationary process, assuming it is linear in martingale differences, but without any restriction on its spectrum. We apply the result to kernel nonparametric fixed-design regression, giving a single central limit theorem which indicates how error spectral behavior at only zero frequency influences the asymptotic distribution and covers long-range, short-range and negative dependence. We show how the regression estimates can be Studentized in the absence of previous knowledge of which form of dependence pertains, and show also that a simpler Studentization is possible when long-range dependence can be taken for granted.

159 citations


Posted Content
TL;DR: In this paper, a central limit theorem is given for certain weighted partial sums of a covariance stationary process, assuming it is linear in martingale differences, but without any restriction on its spectrum.
Abstract: A central limit theorem is given for certain weighted partial sums of a covariance stationary process, assuming it is linear in martingale differences, but without any restriction on its spectrum. We apply the result to kernel nonparametric fixed-design regression, giving a single central limit theorem which indicates how error spectral behavior at only zero frequency influences the asymptotic distribution and covers long-range, short-range and negative dependence. We show how the regression estimates can be Studentized in the absence of previous knowledge of which form of dependence pertains, and show also that a simpler Studentization is possible when long-range dependence can be taken for granted.

149 citations


Journal ArticleDOI
TL;DR: In this paper, a general theory for the construction of confidence intervals or regions in the context of heteroskedastic-dependent data is presented, where the sampling distribution of a statistic based on the values of the statistic computed over smaller subsets of the data is used.

132 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the asymptotic behavior of the Gaussian limit law for a random ergodic sequence of matrices with positive entries in a given set of columns and rows.
Abstract: Let $S$ be the set of $q \times q$ matrices with positive entries, such that each column and each row contains a strictly positive element, and denote by $S^\circ$ the subset of these matrices, all entries of which are strictly positive. Consider a random ergodic sequence $(X_n)_{n \geq1}$ in $S$. The aim of this paper is to describe the asymptotic behavior of the random products $X^{(n)} =X_n \ldots X _1, n\geq 1$ under the main hypothesis $P(\Bigcup_{n\geq 1}[X^{(n)}\in S^\circ])>0$. We first study the behavior “in direction” of row and column vectors of $X^{(n)}$. Then, adding a moment condition, we prove a law of large numbers for the entries and lengths of these vectors and also for the spectral radius of $X^{(n)}$ . Under the mixing hypotheses that are usual in the case of sums of real random variables, we get a central limit theorem for the previous quantities. The variance of the Gaussian limit law is strictly positive except when $(X^{(n)})_{n\geq 1}$ is tight. This tightness property is fully studied when the $X_n, n\geq 1$, are independent.

129 citations


Journal ArticleDOI
TL;DR: In this paper, the authors apply the recently introduced method of hermitization to study in the large $N$ limit non-hermitean random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions.
Abstract: We apply the recently introduced method of hermitization to study in the large $N$ limit non-hermitean random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions, thus extending the recent Gaussian non-hermitean literature. We develop the general formalism for calculating the Green's function and averaged density of eigenvalues, which may be thought of as the non-hermitean analog of the method due to Brezin, Itzykson, Parisi and Zuber for analyzing hermitean non-Gaussian random matrices. We obtain an explicit algebraic equation for the integrated density of eigenvalues. A somewhat surprising result of that equation is that the shape of the eigenvalue distribution in the complex plane is either a disk or an annulus. As a concrete example, we analyze the quartic ensemble and study the phase transition from a disk shaped eigenvalue distribution to an annular distribution. Finally, we apply the method of hermitization to develop the addition formalism for free non-hermitean random variables. We use this formalism to state and prove a non-abelian non-hermitean version of the central limit theorem.

122 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider the nonlinear regression model yx gx x, where gx is an unknown function and x is the noise, and fix the design points x and get ni,
Abstract: iences, mixing sequences or associated sequences. The results are important in analyzing the asymptotical properties of some estimators as well as of linear processes. �4 1. Introduction. Let be a centered sequence of random variables k �4 and let a ,1 i n be a triangular array of numbers. Many statistical ni procedures produce estimators of the type n 1.1 S a . Ž. Ý nn i i i1 To give an example let us consider the nonlinear regression model yx gx x , Ž. Ž . Ž . Ž. Ž . where gx is an unknown function and x is the noise. Now we fix the design points x and we get ni ,

121 citations


Journal ArticleDOI
TL;DR: In this paper, the authors apply the recently introduced method of hermitization to study in the large N limit non-hermitian random matrices that are drawn from a large class of circularly symmetric non-gaussian probability distributions.

116 citations


Journal ArticleDOI
TL;DR: In this article, a central limit theorem for near-epoch-dependent random variables is presented for the case of triangular arrays of mixingale and near-EPO-dependent variables.
Abstract: This paper presents central limit theorems for triangular arrays of mixingale and near-epoch-dependent random variables. The central limit theorem for near-epoch-dependent random variables improves results from the literature in various respects. The approach is to define a suitable Bernstein blocking scheme and apply a martingale difference central limit theorem, which in combination with weak dependence conditions renders the result. The most important application of this central limit theorem is the improvement of the conditions that have to be imposed for asymptotic normality of minimization estimators.

Journal ArticleDOI
TL;DR: Second-order noiseless source coding theorems for the deviation of the codeword lengths from the entropy are formulated and a "one-sided central limit theorem" and a law of the iterated logarithm are proved.
Abstract: Shannon's celebrated source coding theorem can be viewed as a "one-sided law of large numbers". We formulate second-order noiseless source coding theorems for the deviation of the codeword lengths from the entropy. For a class of sources that includes Markov chains we prove a "one-sided central limit theorem" and a law of the iterated logarithm.

Journal ArticleDOI
Emmanuel Rio1
TL;DR: In this article, the authors extended the Lindeberg method for the central limit theorem to strongly mixing sequences and obtained a generalization of Doukhan, Massart, and Rio to nonstationary strongly mixing triangular arrays, and provided estimates of the Levy distance between the distribution of the normalized sum and the standard normal.
Abstract: We extend the Lindeberg method for the central limit theorem to strongly mixing sequences. Here we obtain a generalization of the central limit theorem of Doukhan, Massart and Rio to nonstationary strongly mixing triangular arrays. The method also provides estimates of the Levy distance between the distribution of the normalized sum and the standard normal.

Journal ArticleDOI
TL;DR: In this article, the authors derived general expressions for the law, first moment, and probability generating function of Kn(a), mentioning examples where evaluations can be given, and obtained limit laws for top end spacings with j fixed.
Abstract: summary Recent work has considered properties of the number of observations Xj, independently drawn from a discrete law, which equal the sample maximum X(n) The natural analogue for continuous laws is the number Kn(a) of observations in the interval (X(n)–a, X(n)], where a > 0. This paper derives general expressions for the law, first moment, and probability generating function of Kn(a), mentioning examples where evaluations can be given. It seeks limit laws for n and finds a central limit result when a is fixed and the population law has a finite right extremity. Whenever the population law is attracted to an extremal law, a limit theorem can be found by letting a depend on n in an appropriate manner; thus the limit law is geometric when the extremal law is the Gumbel type. With these results, the paper obtains limit laws for ‘top end’ spacings X(n) - X(n-j) with j fixed.

Journal ArticleDOI
TL;DR: Second-order regular variation is a refinement of the concept of regular variation which is useful for studying rates of convergence in extreme value theory and asymptotic normality of tail estimators as discussed by the authors.

Journal ArticleDOI
TL;DR: In this article, the authors analyzed the difference in the values of an estimating function evaluated at its local maxima on two different subsets of the parameter space, assuming that the true parameter is in each subset but possibly on the boundary.
Abstract: In this paper, we analyze the statistic which is the difference in the values of an estimating function evaluated at its local maxima on two different subsets of the parameter space, assuming that the true parameter is in each subset, but possibly on the boundary. Our results extend known methods by covering a large' class of estimation problems which allow sampling from nonidentically distributed random variables. Specifically, the existence and consistency of the local maximum estimators and asymptotic properties of useful hypothesis tests are obtained under certain law of large number and central limit-type assumptions. Other models covered include those with general log-likelihoods and/or covariates. As an example, the large sample theory of two-way nested random variance components models with covariates is derived from our main results.

Journal ArticleDOI
TL;DR: In this paper, an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale is presented, where the emphasis is put on the use of the Malliavin calculus; the uniform nondegeneracy of the covariance under certain truncation plays an essential role as the Cramer condition did in the case of independent observations.
Abstract: We present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale. The emphasis is put on the use of the Malliavin calculus; the uniform nondegeneracy of the Malliavin covariance under certain truncation plays an essential role as the Cramer condition did in the case of independent observations. Applications to statistics are presented.

Journal ArticleDOI
TL;DR: In this paper, an extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases, under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established.
Abstract: An extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases. Under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established for such processes. A compact law of the iterated logarithm (LIL) is then inferred from the corresponding LIL for the approximating sequence of Gaussian processes. A number of statistical applications of our results are indicated.

Journal ArticleDOI
TL;DR: In this paper, the authors obtained new functional central limit results for Hilbert-valued stochastic approximation procedures for nonparametric recursive generalized method of moment estimators and their functionals in time series econometrics.
Abstract: We obtain new CLTs and FCLTs for Hilbert-valued arrays near epoch dependent on mixing processes, as well as new FCLTs for general Hilbert-valued adapted dependent heterogeneous arrays These theorems are useful in delivering asymptotic distributions for parametric and nonparametric estimators and their functionals in time series econometrics We give three significant applications for near epoch dependent observations: (1) A new CLT for any plug-in estimator of a cumulative distribution function (eg, an empirical cdf, or a cdf estimator based on a kernel density estimator), which can in turn deliver distribution results for many Von Mises functionals; (2) A new limiting distribution result for degenerate U-statistics, which delivers distribution results for Bierens' integrated conditional moment tests; (3) A new functional central limit result for Hilbert-valued stochastic approximation procedures, which delivers distribution results for nonparametric recursive generalized method of moment estimators, including nonparametric adaptive learning models

Book ChapterDOI
01 Jan 1997
TL;DR: Central limit theorems have played a paramount role in probability theory starting with the DeMoivre-Laplace version and culminating with that of Lindeberg-Feller as discussed by the authors.
Abstract: Central limit theorems have played a paramount role in probability theory starting—in the case of independent random variables—with the DeMoivreLaplace version and culminating with that of Lindeberg-Feller. The term “central” refers to the pervasive, although nonunique, role of the normal distribution as a limit of d.f.s of normalized sums of (classically independent) random variables. Central limit theorems also govern various classes of dependent random variables and the cases of martingales and interchangeable random variables will be considered.

Journal ArticleDOI
TL;DR: In this article, the authors generalize their work with a de Finetti-like characterization of the distribution of repeated measures that can be represented with mixtures of likelihoods of independent but not identically distributed random variables.
Abstract: Recently, the problem of characterizing monotone unidimensional latent variable models for binary repeated measures was studied by Ellis and van den Wollenberg and by Junker. We generalize their work with a de Finetti-like characterization of the distribution of repeated measures $\rm X = (X_1, X_2, \dots)$ that can be represented with mixtures of likelihoods of independent but not identically distributed random variables, where the data satisfy a stochastic ordering property with respect to the mixing variable. The random variables $X_j$ may be arbitrary real-valued random variables. We show that the distribution of X can be given a monotone unidimensional latent variable representation that is useful in the sense of Junker if and only if this distribution satisfies conditional association (CA) and a vanishing conditional dependence (VCD) condition, which asserts that finite subsets of the variables in X become independent as we condition on a larger and larger segment of the remaining variables in X. It is also interesting that the mixture representation is in a certain ordinal sense unique, when CA and VCD hold. The characterization theorem extends and simplifies the main result of Junker and generalizes methods of Ellis and van den Wollenberg to a much broader class of models. Exchangeable sequences of binary random variables also satisfy both CA and VCD, as do exchangeable sequences arising as location mixtures. In the same way that de Finetti's theorem provides a path toward justifying standard i.i.d.-mixture components in hierarchical models on the basis of our intuitions about the exchangeability of observations, this theorem justifies one-dimensional latent variable components in hierarchical models, in terms of our intuitions about positive association and redundancy between observations. Because these conditions are on the joint distribution of the observable data X, they may also be used to construct asymptotically power-1 tests for unidimensional latent variable models.

Journal ArticleDOI
TL;DR: In this article, a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions is derived.
Abstract: V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a high-dimensional second order measure are close to each other in most directions. Extending this and a related result in the context of projection pursuit of P. Diaconis and D. Freedman [Dia84], we give for a probability measure \(P\) and a random (a.s.) linear functional \(F\) on a Hilbert space simple sufficient conditions under which most of the one-dimensional images of \(P\) under \(F\) are close to their canonical mixture which turns out to be almost a mixed normal distribution. Using the concept of approximate conditioning we deduce a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions.

Journal ArticleDOI
TL;DR: For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic ΣjN=1 (xj − 〈x) is computed exactly and shown to satisfy a central limit theorem asN → ∞ as mentioned in this paper.
Abstract: For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic ΣjN=1 (xj − 〈x〉) is computed exactly and shown to satisfy a central limit theorem asN → ∞. For the circular random matrix ensemble the p.d.f.’s for the statistics ½ΣjN=1 (θj −π) and − ΣjN=1 log 2 |sinθj/2| are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem asN → ∞.


Journal Article
TL;DR: In this article, the authors give upper and lower bounds for moments of sums of independent random variables (Xk) which satisfy the condition that P (X|k ≥ t) = exp(−Nk(t), where Nk are concave functions.
Abstract: This paper gives upper and lower bounds for moments of sums of independent random variables (Xk) which satisfy the condition that P (|X|k ≥ t) = exp(−Nk(t)), where Nk are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which N(t) = |t| for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapień who have done the same for convex functions N .

Journal ArticleDOI
TL;DR: In this article, the authors considered a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ¯¯¯¯ t�� (x): (t,x)∈ℤn+1}.
Abstract: We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ t (x): (t,x)∈ℤν+1}. The random walk transition probabilities are taken as P(X t +1= y|X t = x,ξ t =η) =P 0( y−x)+ c(y−x;η(x)). We assume that the variables {ξ t (x):(t,x) ∈ℤν+1} are i.i.d., that both P 0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P 0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X t , and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X t and a corresponding correction of order to the C.L.T.. Proofs are based on some new L p estimates for a class of functionals of the field.

Journal ArticleDOI
TL;DR: In this paper, a central limit theorem for the endpoint of the path is proved for the Edwards model in one dimension, and the scaled variance is characterized in terms of the largest eigenvalue of a one-parameter family of differential operators.
Abstract: The Edwards model in one dimension is a transformed path measure for standard Brownian motion discouraging self-intersections. We prove a central limit theorem for the endpoint of the path, extending a law of large numbers proved by Westwater. The scaled variance is characterized in terms of the largest eigenvalue of a one-parameter family of differential operators, introduced and analyzed by van der Hofstad and den Hollander. Interestingly, the scaled variance turns out to be independent of the strength of self-repellence and to be strictly smaller than one (the value for free Brownian motion).

Journal ArticleDOI
TL;DR: In this article, the authors consider random Hermitian matrices made of complex or real M×N rectangular blocks, where the blocks are drawn from various ensembles, and study the eigenvalue distribution of these matrices to leading order in the large-N, M limit in which the "rectangularity"r=M/N is held fixed.
Abstract: We consider random Hermitian matrices made of complex or realM×N rectangular blocks, where the blocks are drawn from various ensembles. These matrices haveN pairs of opposite real nonvanishing eigenvalues, as well asM−N zero eigenvalues (forM>N). These zero eigenvalues are “kinematical” in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large-N, M limit in which the “rectangularity”r=M/N is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large-N renormalization techniques. In addition to the kinematical δ-function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if |r−1| is held fixed asN→∞, theN nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the nonzero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. Asr→1 the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal thatr→1 drives a crossover to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.

Journal ArticleDOI
TL;DR: In this paper, three classes of strictly stationary, strongly mixing random sequences are constructed, in order to provide further information on the borderline of the central limit theorem for strictly stationary and strongly mixed random sequences.
Abstract: Three classes of strictly stationary, strongly mixing random sequences are constructed, in order to provide further information on the “borderline” of the central limit theorem for strictly stationary, strongly mixing random sequences. In these constructions, a key role is played by quantiles, as in a related construction of Doukhan et al.(11)

Journal ArticleDOI
TL;DR: In this paper, the authors consider the problem of running independent, single-stage simulations to make multiple comparisons of the steady-state means of the different systems, and derive asymptotically valid (as the run lengths of the simulations of the systems tend to infinity) simultaneous confidence intervals for each of the following problems.
Abstract: Suppose that there are $k \geq 2$ different systems (i.e., stochastic processes), where each system has an unknown steady-state mean performance and unknown asymptotic variance. We allow for the asymptotic variances to be unequal and for the distributions of the k systems to be different. We consider the problem of running independent, single-stage simulations to make multiple comparisons of the steady-state means of the different systems. We derive asymptotically valid (as the run lengths of the simulations of the systems tend to infinity) simultaneous confidence intervals for each of the following problems: all pairwise comparisons of means, all contrasts, multiple comparisons with a control and multiple comparisons with the best. Our confidence intervals are based on standardized time series methods, and we establish the asymptotic validity of each under the sole assumption that the stochastic processes representing the simulation output of the different systems satisfy a functional central limit theorem. Although simulation is the context of this paper, the results naturally apply to (asymptotically) stationary time series.