Showing papers on "Central limit theorem published in 2010"

Scaling Limits of Interacting Particle Systems

Abstract: 1. An Introductory Example: Independent Random Walks.- 2. Some Interacting Particle Systems.- 3. Weak Formulations of Local Equilibrium.- 4. Hydrodynamic Equation of Symmetric Simple Exclusion Processes.- 5. An Example of Reversible Gradient System: Symmetric Zero Range Processes.- 6. The Relative Entropy Method.- 7. Hydrodynamic Limit of Reversible Nongradient Systems.- 8. Hydrodynamic Limit of Asymmetric Attractive Processes.- 9. Conservation of Local Equilibrium for Attractive Systems.- 10. Large Deviations from the Hydrodynamic Limit.- 11. Equilibrium Fluctuations of Reversible Dynamics.- Appendices.- 1. Markov Chains on a Countable Space.- 1.1 Discrete Time Markov Chains.- 1.2 Continuous Time Markov Chains.- 1.3 Kolmogorov's Equations, Generators.- 1.4 Invariant Measures, Reversibility and Adjoint Processes.- 1.5 Some Martingales in the Context of Markov Processes.- 1.6 Estimates on the Variance of Additive Functionals of Markov Processes.- 1.7 The Feynman-Kac Formula.- 1.8 Relative Entropy.- 1.9 Entropy and Markov Processes.- 1.10 Dirichlet Form.- 1.11 A Maximal Inequality for Reversible Markov Processes.- 2. The Equivalence of Ensembles, Large Deviation Tools and Weak Solutions of Quasi-Linear Differential Equations.- 2.1 Local Central Limit Theorem and Equivalence of Ensembles.- 2.2 On the Local Central Limit Theorem.- 2.3 Remarks on Large Deviations.- 2.4 Weak Solutions of Nonlinear Parabolic Equations.- 2.5 Entropy Solutions of Quasi-Linear Hyperbolic Equations.- 3. Nongradient Tools: Spectral Gap and Closed Forms.- 3.1 On the Spectrum of Reversible Markov Processes.- 3.2 Spectral Gap for Generalized Exclusion Processes.- 3.4 Closed and Exact Forms.- 3.5 Comments and References.- References.

A Modern Introduction to Probability and Statistics: Understanding Why and How

Abstract: Why probability and statistics?.- Outcomes, events, and probability.- Conditional probability and independence.- Discrete random variables.- Continuous random variables.- Simulation.- Expectation and variance.- Computations with random variables.- Joint distributions and independence.- Covariance and correlation.- More computations with more random variables.- The Poisson process.- The law of large numbers.- The central limit theorem.- Exploratory data analysis: graphical summaries.- Exploratory data analysis: numerical summaries.- Basic statistical models.- The bootstrap.- Unbiased estimators.- Efficiency and mean squared error.- Maximum likelihood.- The method of least squares.- Confidence intervals for the mean.- More on confidence intervals.- Testing hypotheses: essentials.- Testing hypotheses: elaboration.- The t-test.- Comparing two samples.

A History of the Central Limit Theorem: From Classical to Modern Probability Theory

Abstract: This book provides a comprehensive history of one of the most important theorems of probability theory and mathematical statistics from about 1810 to ca. 1950. Taking problems surrounding the central limit theorem as characteristic examples, the development of probability theory from its classical form to its modern, and even postmodern, shape is illuminated. The main topics of the book include: Laplace's normal approximations and its applications to natural sciences; Poisson's modifications; Dirichlet's proof; the central limit theorem in the Cauchy-Bienayme controversy; the development of the hypothesis of elementary errors; the theory of moments and the central limit theorem in Chebyshev's and Markov's work; first steps into modern probability by Lyapunov; contributions by Polya, von Mises, Lindeberg, Levy, Bernshtein, and Cramer during the 1920s; Levy's and Feller's necessary conditions; generalizations toward dependent random variables, nonnormal limit laws, functional limit theorems, and sums of random elements in metric spaces.

Efficient and Exact Sampling of Simple Graphs with Given Arbitrary Degree Sequence

Abstract: Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large , and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.

A quantum central limit theorem for non-equilibrium systems: exact local relaxation of correlated states

Abstract: We prove that quantum many-body systems on a one-dimensional lattice locally relax to Gaussian states under non-equilibrium dynamics generated by a bosonic quadratic Hamiltonian. This is true for a large class of initial states—pure or mixed—which have to satisfy merely weak conditions concerning the decay of correlations. The considered setting is a proven instance of a situation where dynamically evolving closed quantum systems locally appear as if they had truly relaxed, to maximum entropy states for fixed second moments. This furthers the understanding of relaxation in suddenly quenched quantum many-body systems. The proof features a non-commutative central limit theorem for non-i.i.d. random variables, showing convergence to Gaussian characteristic functions, giving rise to trace-norm closeness. We briefly link our findings to the ideas of typicality and concentration of measure.

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–Esseen bounds in Central limit theorems (CLTs) involving multiple Wiener–Ito integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein–Uhlenbeck Levy processes.

Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos

Abstract: 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.

Central and non-central limit theorems for weighted power variations of fractional brownian motion

Abstract: In this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of order q>=2 of the fractional Brownian motion with Hurst parameter H in (0,1), where q is an integer. The central limit holds for 1/(2q) 1-1/(2q), we show the convergence in L^2 to a stochastic integral with respect to the Hermite process of order q.

Fluid limit theorems for stochastic hybrid systems with application to neuron models

Abstract: In this paper we establish limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamics coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation, and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to numerically estimate the impact of channel noise both on frequency and spike timing coding.

Fluid limit theorems for stochastic hybrid systems with application to neuron models

Abstract: This paper establishes limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamic coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to estimate numerically impact of channel noise both on frequency and spike timing coding.

The strong law of large numbers for extended negatively dependent random variables

Abstract: A sequence of random variables is said to be extended negatively dependent (END) if the tails of its finite-dimensional distributions in the lower-left and upper-right corners are dominated by a multiple of the tails of the corresponding finite-dimensional distributions of a sequence of independent random variables with the same marginal distributions. The goal of this paper is to establish the strong law of large numbers for a sequence of END and identically distributed random variables. In doing so we derive some new inequalities of large deviation type for the sums of END and identically distributed random variables being suitably truncated. We also show applications of our main result to risk theory and renewal theory.

A CLT and tight lower bounds for estimating entropy.

Abstract: We prove two new multivariate central limit theorems; the first relates the sum of indepen- dent distributions to the multivariate Gaussian of corresponding mean and covariance, under the earthmover distance matric (also known as the Wasserstein metric). We leverage this central limit theorem to prove a stronger but more specific central limit theorem for “generalized multinomial” distributions—a large class of discrete distributions, parameterized by matrices, that generalize binomial and multinomial distributions, and describe many distributions encountered in computer science. This central limit theorem relates a generalized multinomial distribution to a multivari- ate Gaussian distribution, discretized by rounding to the nearest lattice points. In contrast to the metric of our first central limit theorem, this bound is in terms of statistical distance, which imme- diately implies that any algorithm with input drawn from a generalized multinomial distribution behaves essentially as if the input were drawn from a discretized Gaussian with the same mean and covariance. Such tools in the multivariate setting are rare, and we hope this new tool will be of use to the community. In the second part of the paper, we employ this central limit theorem to establish a lower bound of Ω( n log n ) on the sample complexity of additively estimating the entropy or support size of a distribution (where 1/n is a lower bound on the probability of any element in the domain). Together with the canonical estimator constructed in the companion paper [33], this settles the longstanding open question of the sample complexities of these estimation problems, up to constant factors. In particular, for any constants c1 2 , there is a family of pairs of distributions D,D′ each of whose elements occurs with probability at least 1/n, whose entropies satisfy H(D)−H(D′) > c1, and whose support sizes differ by at least c2n, such that no algorithm on o( n log n ) samples can distinguish D from D ′ with probability greater than 2/3. For the problem of estimating entropy, we also provide a bound on the rate of convergence of an optimal estimator, showing that the sample complexity of estimating entropy to within additive c is Ω ( n c log n ) . The previous lower-bounds on these sample complexities were n/2 √ log , for constant c, from [34]. We explicitly exhibit such a family of pairs of distributions D,D′ via a Laguerre polynomial construction that may be of independent interest.

A backward particle interpretation of Feynman-Kac formulae

Abstract: We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals "on-the-fly" as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We provide uniform convergence results w.r.t. the time horizon parameter as well as functional central limit theorems and exponential concentration estimates, yielding what seems to be the first results of this type for this class of models. We also illustrate these results in the context of filtering of hidden Markov models, as well as in computational physics and imaginary time Schroedinger type partial differential equations, with a special interest in the numerical approximation of the invariant measure associated to h-processes.

On the Upper Bound for the Absolute Constant in the Berry–Esseen Inequality

Abstract: This paper describes the history of the search for unconditional and conditional upper bounds of the absolute constant in the Berry–Esseen inequality for sums of independent identically distributed random variables. Computational procedures are described. New estimates are presented from which it follows that the absolute constant in the classical Berry–Esseen inequality does not exceed 0.5129.

Limit theorems for moving averages of discretized processes plus noise

Abstract: This paper presents some limit theorems for certain functionals of moving averages of semimartingales plus noise which are observed at high frequency. Our method generalizes the pre-averaging approach (see [Bernoulli 15 (2009) 634–658, Stochastic Process. Appl. 119 (2009) 2249–2276]) and provides consistent estimates for various characteristics of general semimartingales. Furthermore, we prove the associated multidimensional (stable) central limit theorems. As expected, we find central limit theorems with a convergence rate n−1/4, if n is the number of observations.

On the Asymptotic Spectrum of Products of Independent Random Matrices.

Abstract: We consider products of independent random matrices with independent entries. The limit distribution of the expected empirical distribution of eigenvalues of such products is computed. Let X (�)

Théorie ergodique des fractions rationnelles sur un corps ultramétrique

Abstract: We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure m_R which reprensents the asymptotic distribution of preimages of non-exceptional point. We show that this measure is exponentially mixing, and satisfies the central limit theorem. We prove some general bounds on the metric entropy of m_R, and on the topological entropy of R. We finally prove that rational maps with vanishing topological entropy have potential good reduction.

Multi-Dimensional Gaussian Fluctuations on the Poisson Space

Abstract: We study multi-dimensional normal approximations on the Poisson space by means of Malliavin calculus, Stein's method and probabilistic interpolations. Our results yield new multi-dimensional central limit theorems for multiple integrals with respect to Poisson measures - thus significantly extending previous works by Peccati, Sole, Taqqu and Utzet. Several explicit examples (including in particular vectors of linear and non-linear functionals of Ornstein-Uhlenbeck Levy processes) are discussed in detail.

Bootstrap percolation on the random graph $G_{n,p}$

Abstract: Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least $r\geq2$ active neighbors become active as well. We study the size $A^*$ of the final active set. The parameters of the model are, besides $r$ (fixed) and $n$ (tending to $\infty$), the size $a=a(n)$ of the initially active set and the probability $p=p(n)$ of the edges in the graph. We show that the model exhibits a sharp phase transition: depending on the parameters of the model, the final size of activation with a high probability is either $n-o(n)$ or it is $o(n)$. We provide a complete description of the phase diagram on the space of the parameters of the model. In particular, we find the phase transition and compute the asymptotics (in probability) for $A^*$; we also prove a central limit theorem for $A^*$ in some ranges. Furthermore, we provide the asymptotics for the number of steps until the process stops.

First passage percolation on random graphs with finite mean degrees

Abstract: We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the so-called hopcount. We analyze the configuration model with degree power-law exponent τ > 2, in which the degrees are assumed to be i.i.d. with a tail distribution which is either of power-law form with exponent τ − 1 > 1, or has even thinner tails (τ = ∞). In this model, the degrees have a finite first moment, while the variance is finite for τ > 3, but infinite for τ ∈ (2, 3). We prove a central limit theorem for the hopcount, with asymptotically equal means and variances equal to α log n, where α ∈ (0, 1) for τ ∈ (2, 3), while α > 1 for τ > 3. Here n denotes the size of the graph. For τ ∈ (2, 3), it is known that the graph distance between two randomly chosen connected vertices is proportional to log log n [25], i.e., distances are ultra small. Thus, the addition of edge weights causes a marked change in the geometry of the network. We further study the weight of the least weight path, and prove convergence in distribution of an appropriately centered version. This study continues the program initiated in [5] of showing that log n is the correct scaling for the hopcount under i.i.d. edge disorder, even if the graph distance between two randomly chosen vertices is of much smaller order. The case of infinite mean degrees (τ ∈ [1, 2)) is studied in [6], where it is proved that the hopcount remains uniformly bounded and converges in distribution.

Moderate deviation analysis of channel coding: Discrete memoryless case

Abstract: Moderate deviation behavior of coding for discrete-memoryless channels is investigated. That is, we consider block codes whose rate converges to the channel capacity from below with increasing block length with a certain rate and examine the best ‘sub-exponential’ decay in the maximal probability of error. We prove that a moderate deviation principle (M.D.P.) holds for all convergence rates between the large deviation and the central limit theorem regimes, under some mild assumptions on the channel. The rate function of the M.D.P. is explicitly characterized.

Stein's Method for Dependent Random Variables Occuring in Statistical Mechanics

Abstract: We develop Stein's method for exchangeable pairs for a rich class of distributional approximations including the Gaussian distributions as well as the non-Gaussian limit distributions. As a consequence we obtain convergence rates in limit theorems of partial sums for certain sequences of dependent, identically distributed random variables which arise naturally in statistical mechanics, in particular in the context of the Curie-Weiss models. Our results include a Berry-Esseen rate in the Central Limit Theorem for the total magnetization in the classical Curie-Weiss model, for high temperatures as well as at the critical temperature, where the Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds as the temperature converges to one and obtain a threshold for the speed of this convergence. Single spin distributions satisfying the Griffiths-Hurst-Sherman (GHS) inequality like models of liquid helium or continuous Curie-Weiss models are considered.

Kinetic limits for waves in a random medium

Abstract: 2 Diffusive limit for a particle in a random flow 7 2.1 Diffusion of a particle in a time-dependent random flow . . . . . . . . . . . . . . . . 7 2.1.1 The central limit theorem, purely time-dependent flows, and diffusion . . . . 8 2.1.2 Using formal asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Random flows with spatial-temporal dependence . . . . . . . . . . . . . . . . 10 2.2 The proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 One and two particles in a random flow with a strong drift . . . . . . . . . . . . . . 21

Mesoscopic fluctuations of the zeta zeros

Abstract: We prove a multidimensional extension of Selberg’s central limit theorem for log ζ, in which non-trivial correlations appear. In particular, this answers a question by Coram and Diaconis about the mesoscopic fluctuations of the zeros of the Riemann zeta function. Similar results are given in the context of random matrices from the unitary group. This shows the correspondence n ↔ log t not only between the dimension of the matrix and the height on the critical line, but also, in a local scale, for small deviations from the critical axis or the unit circle.

Limit theorems for empirical processes of cluster functionals

Abstract: Let (X-n, i) 1 <= i <= n,m is an element of N be a triangular array of row-wise stationary R-d-valued random variables. We use a "blocks method" to define clusters of extreme values: the rows of (X-n, i) are divided into m(n) blocks (Y-n, j), and if a block contains at least one extreme value, the block is considered to contain a cluster. The cluster starts at the first extreme value in the block and ends at the last one. The main results are uniform central limit theorems for empirical processes Z(n)(f) := 1/root nv(n) Sigma(mn)(j=1) (f(Y-n,Y- j) - Ef(Y-n,Y- j)), for v(n) = P{X-n,X- i not equal 0} and f belonging to classes of cluster functionals, that is, functions of the blocks Y-n,Y- j which only depend on the cluster values and which are equal to 0 if Y-n,Y- j does not contain a cluster. Conditions for finite-dimensional convergence include beta-mixing, suitable Lindeberg conditions and convergence of covariances. To obtain full uniform convergence, we use either "bracketing entropy" or bounds on covering numbers with respect to a random semi-metric. The latter makes it possible to bring the powerful Vapnik-Cervonenkis theory to bear. Applications include multivariate tail empirical processes and empirical processes of cluster values and of order statistics in clusters. Although our main field of applications is the analysis of extreme values, the theory can be applied more generally to rare events occurring, for example, in nonparametric curve estimation.

On the ergodicity of the adaptive Metropolis algorithm on unbounded domains

Abstract: This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman, and Tamminen (9), for target distributions with a non-compact support. The conditions ensuring a strong law of large numbers and a central limit theorem require that the tails of the target density decay super-exponentially and have regular contours. The result is based on the ergodicity of an auxiliary process that is sequentially constrained to feasible adaptation sets, and independent estimates of the growth rate of the AM chain and the corresponding geometric drift constants. The ergodicity result of the constrained process is obtained through a modification of the approach due to Andrieu and Moulines (1).