Showing papers on "Central limit theorem published in 2008"

Asymptotic Theory of Statistics and Probability

Abstract: Basic Convergence Concepts and Theorems.- Metrics, Information Theory, Convergence, and Poisson Approximations.- More General Weak and Strong Laws and the Delta Theorem.- Transformations.- More General Central Limit Theorems.- Moment Convergence and Uniform Integrability.- Sample Percentiles and Order Statistics.- Sample Extremes.- Central Limit Theorems for Dependent Sequences.- Central Limit Theorem for Markov Chains.- Accuracy of Central Limit Theorems.- Invariance Principles.- Edgeworth Expansions and Cumulants.- Saddlepoint Approximations.- U-statistics.- Maximum Likelihood Estimates.- M Estimates.- The Trimmed Mean.- Multivariate Location Parameter and Multivariate Medians.- Bayes Procedures and Posterior Distributions.- Testing Problems.- Asymptotic Efficiency in Testing.- Some General Large-Deviation Results.- Classical Nonparametrics.- Two-Sample Problems.- Goodness of Fit.- Chi-square Tests for Goodness of Fit.- Goodness of Fit with Estimated Parameters.- The Bootstrap.- Jackknife.- Permutation Tests.- Density Estimation.- Mixture Models and Nonparametric Deconvolution.- High-Dimensional Inference and False Discovery.- A Collection of Inequalities in Probability, Linear Algebra, and Analysis.

Probability: A Graduate Course

Abstract: Preface to the First Edition.- Preface to the Second Edition.- Outline of Contents.- Notation and Symbols.- Introductory Measure Theory.- Random Variables.- Inequalities.- Characteristic Functions.- Convergence.- The Law of Large Numbers.- The Central Limit Theorem.- The Law of the Iterated Logarithm.- Limited Theorems.- Martingales.- Some Useful Mathematics.- References.- Index.

Central limit theorems for eigenvalues in a spiked population model

Abstract: In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope with empirical findings on various data sets. The question is to quantify the effect of the perturbation caused by the spike eigenvalues. A recent work by Baik and Silverstein establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. This paper establishes the limiting distributions of these extreme sample eigenvalues. As another important result of the paper, we provide a central limit theorem on random sesquilinear forms.

A New Central Limit Theorem under Sublinear Expectations

Abstract: We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal-distribution in the sense that mean-uncertainty can be also described. W present our new result of central limit theorem under sublinear expectation. This theorem can be also regarded as a generalization of the law of large number in the case of mean-uncertainty.

Limit theorems for weighted samples with applications to sequential Monte Carlo methods

Abstract: In the last decade, sequential Monte Carlo methods (SMC) emerged as a key tool in computational statistics [see, e.g., Sequential Monte Carlo Methods in Practice (2001) Springer, New York, Monte Carlo Strategies in Scientific Computing (2001) Springer, New York, Complex Stochastic Systems (2001) 109–173]. These algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles, which are generated recursively. Despite many theoretical advances [see, e.g., J. Roy. Statist. Soc. Ser. B 63 (2001) 127–146, Ann. Statist. 33 (2005) 1983–2021, Feynman–Kac Formulae. Genealogical and Interacting Particle Systems with Applications (2004) Springer, Ann. Statist. 32 (2004) 2385–2411], the large-sample theory of these approximations remains a question of central interest. In this paper we establish a law of large numbers and a central limit theorem as the number of particles gets large. We introduce the concepts of weighted sample consistency and asymptotic normality, and derive conditions under which the transformations of the weighted sample used in the SMC algorithm preserve these properties. To illustrate our findings, we analyze SMC algorithms to approximate the filtering distribution in state-space models. We show how our techniques allow to relax restrictive technical conditions used in previously reported works and provide grounds to analyze more sophisticated sequential sampling strategies, including branching, resampling at randomly selected times, and so on.

A new method of normal approximation

Abstract: 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.

Introduction to the Theory of Lévy Flights

Abstract: Levy flights, also referred to as Levy motion, stand for a class of non-Gaussian random processes whose stationary increments are distributed according to a Levy stable distribution originally studied by French mathematician Paul Pierre Levy. Levy stable laws are important for three fundamental properties: (i) similar to the Gaussian law, Levy stable laws form the basin of attraction for sums of random variables. This follows from the theory of stable laws, according to which a generalized central limit theorem exists for random variables with diverging variance. The Gaussian distribution is located at the boundary of the basin of attraction of stable laws; (ii) the probability density functions of Levy stable laws decay in asymptotic power-law form with diverging variance and thus appear naturally in the description of many fluctuation processes with largely scattering statistics characterized by bursts or large outliers; (iii) Levy flights are statistically self-affine, a property catering for the description of random fractal processes. Levy stable laws appear as statistical description for a broad class of processes in physical, chemical, biological, geophysical, or financial contexts, among others. We here review the fundamental properties of Levy flights and their underlying stable laws. Particular emphasis lies on recent developments such as the first passage time and leapover properties of Levy flights, as well as the behavior of Levy flights in external fields. These properties are discussed on the basis of analytical and numerical solutions of fractional kinetic equations as well as numerical solution of the Langevin equation with white Levy noise.

Particle filters for partially observed diffusions

Abstract: In this paper we introduce novel particle filters for a class of partially-observed continuous-time dynamic models where the signal is given by a multivariate diffusion process. We consider a variety of observation schemes, including diffusion observed with error, observation of a subset of the components of the multivariate diffusion and arrival times of a Poisson process whose intensity is a known function of the diffusion (Cox process). Unlike currently available methods, our particle filters do not require approximations of the transition and/or the observation density using time-discretisations. Instead, they build on recent methodology for the exact simulation of the diffusion process and the unbiased estimation of the transition density as described in Beskos et al. (2006). In particular, we introduce the Generalised Poisson Estimator, which generalises the Poisson Estimator of Beskos et al. (2006). Thus, our filters avoid the systematic biases caused by time-discretisations and they have significant computational advantages over alternative continuous-time filters. These advantages are supported theoretically by a central limit theorem.

Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mapping

Abstract: The emphasis of this course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. We choose to show their effectiveness and to describe the theory for two large families of maps. The first chapter deals with holomorphic endomorphisms of the projective space P^k. We establish the first properties and give several constructions for the Green currents and the equilibrium measure \mu. The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems and establish ergodic properties of \mu: K-mixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. Finally, we study the entropy, the Lyapounov exponents and the dimension of \mu. The second chapter develops the theory of polynomial-like maps in higher dimension. We introduce the dynamical degrees and construct the equilibrium measure \mu of maximal entropy. Then, under a natural assumption, we prove equidistribution properties of points and various statistical properties of the measure \mu. The assumption is stable under small pertubations on the map. We also study the dimension of \mu, the Lyapounov exponents and their variation. Our aim is to get a self-contained text that requires only a minimal background. In order to help the reader, an appendix gives the basics on p.s.h. functions, positive closed currents and super-potentials on projective spaces. Some exercises are proposed and an extensive bibliography is given.

Central limit theorems for Gaussian polytopes

Abstract: Choose n random, independent points in R d according to the standard normal distribution. Their convex hull K n is the Gaussian random polytope. We prove that the volume and the number of faces of K n satisfy the central limit theorem, settling a well-known conjecture in the field.

Stein's method and normal approximation of Poisson functionals

Abstract: We combine Stein's method with a version of Malliavin calculus on the Poisson space. As a result, we obtain explicit Berry-Ess\'een bounds in Central Limit Theorems (CLTs) involving multiple Wiener-It\^o integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein-Uhlenbeck L\'evy processes.

Normal and Anomalous Diusion: A Tutorial

Abstract: The purpose of this tutorial is to introduce the main concepts behind normal and anomalous diusion. Starting from simple, but well known experiments, a series of mathematical modeling tools are introduced, and the relation between them is made clear. First, we show how Brownian motion can be understood in terms of a simple random walk model. Normal diusion is then treated (i) through formalizing the random walk model and deriving a classical diusion equation, (ii) by using Fick’s law that leads again to the same diusion equation, and (iii) by using a stochastic dieren tial equation for the particle dynamics (the Langevin equation), which allows to determine the mean square displacement of particles. (iv) We discuss normal diusion from the point of view of probability theory, applying the Central Limit Theorem to the random walk problem, and (v) we introduce the more general Fokker-Planck equation for diusion that includes also advection. We turn then to anomalous diusion, discussing rst its formal characteristics, and proceeding to Continuous Time Random Walk (CTRW) as a model for anomalous diusion. It is shown how CTRW can be treated formally, the importance of probability distributions of the Levy type is explained, and we discuss the relation of CTRW to fractional diusion equations and show how the latter can be derived from the CTRW equations. Last, we demonstrate how a general diusion equation can be derived for Hamiltonian systems, and we conclude this tutorial with a few recent applications of the above theories in laboratory and astrophysical plasmas.

Estimating the Effective Sample Size to Select Independent Measurements in a Reverberation Chamber

Abstract: In reverberation chambers (RCs), measurements are usually performed by changing the boundary conditions using a mode stirrer. The major difficulty is to select uncorrelated samples in order to make a statistical analysis of the data. Furthermore, the knowledge of the number of independent samples is of crucial importance to assess the measurement accuracy. To evaluate whether measured data are independent, the conventional method compares the autocorrelation function (ACF) with the critical value 0.37. However, this criterion is generally not appropriate because the ACF probability density function (pdf) depends strongly on the sample size. For a measurement series of length N, the effective sample size (ESS) is defined as the number N' < N of independent samples, which would provide the same information as the N-size sample. This paper aims to provide a new method based on autoregressive (AR) models and the central limit theorem (CLT) in the case of dependent data, for estimating the ESS. The proposed method is easy to implement since it requires only the knowledge of simple statistical parameters. Moreover, it provides useful guidelines to assess the maximum number of independent samples available with the mode stirrer. Experimental results are in good agreement with the theoretical models, either for the electric field or the received power.

Random integrals and correctors in homogenization

Abstract: This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well-known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well-known that the corrector to homogenization, i.e., the difference between the random solution and the homogenized solution, converges in distribution to a Gaussian process when the correlations in the random medium are sufficiently short-range. Moreover, the limiting process may be written as a stochastic integral with respect to standard Brownian motion. We generalize the result to a large class of processes with long-range correlations. In this setting, the corrector also converges to a Gaussian random process, which has an interpretation as a stochastic integral with respect to fractional Brownian motion. Moreover, we show that the longer the range of the correlations, the larger is the amplitude of the corrector. Derivations are based on a careful analysis of random oscillatory integrals of processes with long-range correlations. We also make use of the explicit expressions for the solutions to the one-dimensional elliptic equation.

Random walk in Markovian environment

Abstract: We prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on $\Z^d$. We assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is Markovian with strong spatial and temporal mixing properties.

Positively and negatively excited random walks on integers, with branching processes

Abstract: We consider excited random walks on the integers with a bounded number of i.i.d. cookies per site which may induce drifts both to the left and to the right. We extend the criteria for recurrence and transience by M. Zerner and for positivity of speed by A.-L. Basdevant and A. Singh to this case and also prove an annealed central limit theorem. The proofs are based on results from the literature concerning branching processes with migration and make use of a certain renewal structure.

Asymptotic normality of the k-core in random graphs

Abstract: We study the k-core of a random (multi)graph on n vertices with a given degree sequence. In our previous paper [Random Structures Algorithms 30 (2007) 50–62] we used properties of empirical distributions of independent random variables to give a simple proof of the fact that the size of the giant k-core obeys a law of large numbers as n→∞. Here we develop the method further and show that the fluctuations around the deterministic limit converge to a Gaussian law above and near the threshold, and to a non-normal law at the threshold. Further, we determine precisely the location of the phase transition window for the emergence of a giant k-core. Hence, we deduce corresponding results for the k-core in G(n, p) and G(n, m).

From ballistic to diffusive behavior in periodic potentials

Abstract: The long-time/large-scale, small-friction asymptotic for the one dimensional Langevin equation with a periodic potential is studied in this paper. It is shown that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We prove that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. We show that the same result is valid for a whole one parameter family of space/time rescalings. The proofs of our main results are based on some novel estimates on the resolvent of a hypoelliptic operator.

Strictly and asymptotically scale invariant probabilistic models of N correlated binary random variables having q-Gaussians as N → ∞ limiting distributions

Abstract: The celebrated Leibnitz triangle has a remarkable property, namely that each of its elements equals the sum of its south-west and south-east neighbors. In probabilistic terms, this corresponds to a specific form of correlation of N equally probable binary variables which satisfy scale invariance. Indeed, the marginal probabilities of the N-system precisely coincide with the joint probabilities of the (N−1)-system. On the other hand, the non-additive entropy , which grounds non-extensive statistical mechanics, is, under appropriate constraints, extremized by the (q-Gaussian) distribution (q<3; ). These distributions also result, as attractors, from a generalized central limit theorem for random variables which have a finite generalized variance, and are correlated in a specific way called q-independence. In order to provide physical enlightenment as regards this concept, we introduce here three types of asymptotically scale invariant probabilistic models with binary random variables, namely (i) a family, characterized by an index ν = 1,2,3,..., unifying the Leibnitz triangle (ν = 1) and the case of independent variables (); (ii) two slightly different discretizations of q-Gaussians; (iii) a special family, characterized by the parameter χ, which generalizes the usual case of independent variables (recovered for χ = 1/2). Models (i) and (iii) are in fact strictly scale invariant. For models (i), we analytically show that the N → ∞ probability distribution is a q-Gaussian with q = (ν−2)/(ν−1). Models (ii) approach q-Gaussians by construction, and we numerically show that they do so with asymptotic scale invariance. Models (iii), like two other strictly scale invariant models recently discussed by Hilhorst and Schehr, approach instead limiting distributions which are not q-Gaussians. The scenario which emerges is that asymptotic (or even strict) scale invariance is not sufficient but it might be necessary for having strict (or asymptotic) q-independence, which, in turn, mandates q-Gaussian attractors.

On the infimum convolution inequality

Abstract: In the paper we study the infimum convolution inequalites. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC-inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure. In particular, we show the optimal IC-inequality for product log-concave measures and for uniform measures on the l_p^n balls. Such an optimal inequality implies, for a given measure, in particular the Central Limit Theorem of Klartag and the tail estimates of Paouris.

A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model

Abstract: We give a closer look at the Central Limit Theorem (CLT) behavior in quasi-stationary states of the Hamiltonian Mean Field model, a paradigmatic one for long-range-interacting classical many-body systems. We present new calculations which show that, following their time evolution, we can observe and classify three kinds of long-standing quasi-stationary states (QSS) with different correlations. The frequency of occurrence of each class depends on the size of the system. The different microscopic nature of the QSS leads to different dynamical correlations and therefore to different results for the observed CLT behavior.

The characteristic polynomial of a random unitary matrix: A probabilistic approach

Abstract: In this paper, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin Fourier transform of such a random polynomial, first obtained by Keating and Snaith in (7), using a simple recursion formula, and from there we are able to obtain the joint law of its radial and an- gular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of in- dependent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in (7) is now obtained from the classical central limit theorems of Probability Theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm type results.

Global fluctuations in physical systems: a subtle interplay between sum and extreme value statistics

Abstract: Fluctuations of global additive quantities, like total energy or magnetization for instance, can in principle be described by statistics of sums of (possibly correlated) random variables. Yet, it turns out that extreme values (the largest value among a set of random variables) may also play a role in the statistics of global quantities, in a direct or indirect way. This review discusses different connections that may appear between problems of sums and of extreme values of random variables, and emphasizes physical situations in which such connections are relevant. Along this line of thought, standard convergence theorems for sums and extreme values of independent and identically distributed random variables are recalled, and some rigorous results as well as more heuristic reasonings are presented for correlated or non-identically distributed random variables. More specifically, the role of extreme values within sums of broadly distributed variables is addressed, and a general mapping between extreme values and sums is presented, allowing us to identify a class of correlated random variables whose sum follows (generalized) extreme value distributions. Possible applications of this specific class of random variables are illustrated on the example of two simple physical models. A few extensions to other related classes of random variables sharing similar qualitative properties are also briefly discussed, in connection with the so-called BHP distribution.

Ordered Random Walks

Abstract: We construct the conditional version of $k$ independent and identically distributed random walks on $R$ given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random walks, the discrete variant of Dyson's Brownian motions, which have been considered yet only for nearest-neighbor walks on the lattice. Our only assumptions are moment conditions on the steps and the validity of the local central limit theorem. The conditional process is constructed as a Doob $h$-transform with some positive regular function $V$ that is strongly related with the Vandermonde determinant and reduces to that function for simple random walk. Furthermore, we prove an invariance principle, i.e., a functional limit theorem towards Dyson's Brownian motions, the continuous analogue.

Limit Theorems for Moving Averages of Discretized Processes Plus Noise

Abstract: This paper presents some limit theorems for certain functionals of moving averages of semi-martingales plus noise, which are observed at high frequency. Our method generalizes the pre-averaging approach (see [13],[11]) and provides consistent estimates for various characteristics of general semi-martingales. Furthermore, we prove the associated multidimensional (stable) central limit theorems. As expected, we find central limit theorems with a convergence rate n1=4, if n is the number of observations.