Showing papers on "Central limit theorem published in 2009"

Level sets and extrema of random processes and fields

Abstract: Introduction. Reading diagram. Chapter 1: Classical results on the regularity of the paths. 1. Kolmogorov's Extension Theorem. 2. Reminder on the Normal Distribution. 3. 0-1 law for Gaussian processes. 4. Regularity of the paths. Exercises. Chapter 2: Basic Inequalities for Gaussian Processes. 1. Slepian type inequalities. 2. Ehrhard's inequality. 3. Gaussian isoperimetric inequality. 4. Inequalities for the tails of the distribution of the supremum. 5. Dudley's inequality. Exercises. Chapter 3: Crossings and Rice formulas for 1-dimensional parameter processes. 1. Rice Formulas. 2. Variants and Examples. Exercises. Chapter 4: Some Statistical Applications. 1. Elementary bounds for P{M > u}. 2. More detailed computation of the first two moments. 3. Maximum of the absolute value. 4. Application to quantitative gene detection. 5. Mixtures of Gaussian distributions. Exercises. Chapter 5: The Rice Series. 1. The Rice Series. 2. Computation of Moments. 3. Numerical aspects of Rice Series. 4. Processes with Continuous Paths. Chapter 6: Rice formulas for random fields. 1. Random fields from Rd to Rd. 2. Random fields from Rd to Rd!, d> d!. Exercises. Chapter 7: Regularity of the Distribution of the Maximum. 1. The implicit formula for the density of the maximum. 2. One parameter processes. 3. Continuity of the density of the maximum of random fields. Exercises. Chapter 8: The tail of the distribution of the maximum. 1. One-dimensional parameter: asymptotic behavior of the derivatives of FM. 2. An Application to Unbounded Processes. 3. A general bound for pM. 4. Computing p(x) for stationary isotropic Gaussian fields. 5. Asymptotics as x! +". 6. Examples. Exercises. Chapter 9: The record method. 1. Smooth processes with one dimensional parameter. 2. Non-smooth Gaussian processes. 3. Two-parameter Gaussian processes. Exercises. Chapter 10: Asymptotic methods for infinite time horizon. 1. Poisson character of "high" up-crossings. 2. Central limit theorem for non-linear functionals. Exercises. Chapter 11: Geometric characteristics of random sea-waves. 1. Gaussian model for infinitely deep sea. 2. Some geometric characteristics of waves. 3. Level curves, crests and velocities for space waves. 4. Real Data. 5. Generalizations of the Gaussian model. Exercises. Chapter 12: Systems of random equations. 1. The Shub-Smale model. 2. More general models. 3. Non-centered systems (smoothed analysis). 4. Systems having a law invariant under orthogonal transformations and translations. Chapter 13: Random fields and condition numbers of random matrices. 1. Condition numbers of non-Gaussian matrices. 2. Condition numbers of centered Gaussian matrices. 3. Non-centered Gaussian matrices. Notations. References.

Zeros of Gaussian Analytic Functions and Determinantal Point Processes

Abstract: The book examines in some depth two important classes of point processes, determinantal processes and 'Gaussian zeros', i.e., zeros of random analytic functions with Gaussian coefficients. These processes share a property of 'point-repulsion', where distinct points are less likely to fall close to each other than in processes, such as the Poisson process, that arise from independent sampling. Nevertheless, the treatment in the book emphasizes the use of independence: for random power series, the independence of coefficients is key; for determinantal processes, the number of points in a domain is a sum of independent indicators, and this yields a satisfying explanation of the central limit theorem (CLT) for this point count. Another unifying theme of the book is invariance of considered point processes under natural transformation groups. The book strives for balance between general theory and concrete examples. On the one hand, it presents a primer on modern techniques on the interface of probability and analysis. On the other hand, a wealth of determinantal processes of intrinsic interest are analyzed; these arise from random spanning trees and eigenvalues of random matrices, as well as from special power series with determinantal zeros. The material in the book formed the basis of a graduate course given at the IAS-Park City Summer School in 2007; the only background knowledge assumed can be acquired in first-year graduate courses in analysis and probability.

Stein's method on Wiener chaos

Abstract: 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.

Central limit theorem for linear eigenvalue statistics of random matrices with independent entries

Abstract: We consider n x n real symmetric and Hermitian Wigner random matrices n ―1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ―1 X*X with independent entries of m x n matrix X. Assuming first that the 4th cumulant (excess) κ 4 of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, m/n → c ∈ [0, oo) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C 5 ). This is done by using a simple "interpolation trick" from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially ℂ 5 test function. Here the variance of statistics contains an additional term proportional to κ 4 . The proofs of all limit theorems follow essentially the same scheme.

Fluctuations of eigenvalues and second order Poincaré inequalities

Abstract: Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems The proofs of such results are usually rather difficult, involving hard computations specific to the model in question In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments In the process, we introduce a notion of ‘second order Poincare inequalities’: just as ordinary Poincare inequalities give variance bounds, second order Poincare inequalities give central limit theorems The proof of the main result employs Stein’s method of normal approximation A number of examples are worked out, some of which are new One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices

Corrections to LRT on Large Dimensional Covariance Matrix by RMT

Abstract: In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension is large compared to the sample size. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F-matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate p (around 20) and high dimension, while the traditional LR tests with chi-square approximation fails. Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.

Corrections to LRT on large-dimensional covariance matrix by RMT

Abstract: In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension p is large compared to the sample size n. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F -matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate p (around 20) and high dimension, while the traditional LR tests with χ 2 approximation fails. Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.

Maximum Entropy Bootstrap for Time Series: The meboot R Package

Abstract: The maximum entropy bootstrap is an algorithm that creates an ensemble for time series inference. Stationarity is not required and the ensemble satisfies the ergodic theorem and the central limit theorem. The meboot R package implements such algorithm. This document introduces the procedure and illustrates its scope by means of several guided applications.

Central limit theorem for linear eigenvalue statistics of ranfom matrices with independent entries.

Abstract: We consider n x n real symmetric and Hermitian Wigner random matrices n ―1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ―1 X*X with independent entries of m x n matrix X. Assuming first that the 4th cumulant (excess) κ 4 of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, m/n → c ∈ [0, oo) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C 5 ). This is done by using a simple "interpolation trick" from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially ℂ 5 test function. Here the variance of statistics contains an additional term proportional to κ 4 . The proofs of all limit theorems follow essentially the same scheme.

Asymptotics for spherical needlets

Abstract: We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasi-exponentially decaying tails. We show that, for random fields on the sphere, the needlet coefficients are asymptotically uncorrelated for any fixed angular distance. This property is used to derive CLT and functional CLT convergence results for polynomial functionals of the needlet coefficients: here the asymptotic theory is considered in the high-frequency sense. Our proposals emerge from strong empirical motivations, especially in connection with the analysis of cosmological data sets.

Universality of rank-ordering distributions in the arts and sciences.

Abstract: Searching for generic behaviors has been one of the driving forces leading to a deep understanding and classification of diverse phenomena. Usually a starting point is the development of a phenomenology based on observations. Such is the case for power law distributions encountered in a wealth of situations coming from physics, geophysics, biology, lexicography as well as social and financial networks. This finding is however restricted to a range of values outside of which finite size corrections are often invoked. Here we uncover a universal behavior of the way in which elements of a system are distributed according to their rank with respect to a given property, valid for the full range of values, regardless of whether or not a power law has previously been suggested. We propose a two parameter functional form for these rank-ordered distributions that gives excellent fits to an impressive amount of very diverse phenomena, coming from the arts, social and natural sciences. It is a discrete version of a generalized beta distribution, given by f(r) = A(N+1-r)b/ra, where r is the rank, N its maximum value, A the normalization constant and (a, b) two fitting exponents. Prompted by our genetic sequence observations we present a growth probabilistic model incorporating mutation-duplication features that generates data complying with this distribution. The competition between permanence and change appears to be a relevant, though not necessary feature. Additionally, our observations mainly of social phenomena suggest that a multifactorial quality resulting from the convergence of several heterogeneous underlying processes is an important feature. We also explore the significance of the distribution parameters and their classifying potential. The ubiquity of our findings suggests that there must be a fundamental underlying explanation, most probably of a statistical nature, such as an appropriate central limit theorem formulation.

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 relate our findings to ideas of typicality and concentration of measure.

Closer look at time averages of the logistic map at the edge of chaos.

Abstract: The probability distribution of sums of iterates of the logistic map at the edge of chaos has been recently shown [U. Tirnakli et al., Phys. Rev. E 75, 040106(R) (2007)] to be numerically consistent with a $q$-Gaussian, the distribution which---under appropriate constraints---maximizes the nonadditive entropy ${S}_{q}$, which is the basis of nonextensive statistical mechanics. This analysis was based on a study of the tails of the distribution. We now check the entire distribution, in particular, its central part. This is important in view of a recent $q$ generalization of the central limit theorem, which states that for certain classes of strongly correlated random variables the rescaled sum approaches a $q$-Gaussian limit distribution. We numerically investigate for the logistic map with a parameter in a small vicinity of the critical point under which conditions there is convergence to a $q$-Gaussian both in the central region and in the tail region and find a scaling law involving the Feigenbaum constant $\ensuremath{\delta}$. Our results are consistent with a large number of already available analytical and numerical evidences that the edge of chaos is well described in terms of the entropy ${S}_{q}$ and its associated concepts.

Upper bounds for minimal distances in the central limit theorem

Abstract: Nous obtenons des majorations des distances minimales dans le theoreme limite central pour les suites de variables aleatoires reelles independantes.

Locally stationary long memory estimation

Abstract: There exists a wide literature on modelling strongly dependent time series using a longmemory parameter d, including more recent work on semiparametric wavelet estimation. As a generalization of these latter approaches, in this work we allow the long-memory parameter d to be varying over time. We embed our approach into the framework of locally stationary processes. We show weak consistency and a central limit theorem for our log-regression wavelet estimator of the time-dependent d in a Gaussian context. Both simulations and a real data example complete our work on providing a fairly general approach.

Computing VaR and CVaR using Stochastic Approximation and Adaptive Unconstrained Importance Sampling

Abstract: Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR) are two risk measures which are widely used in the practice of risk management. This paper deals with the problem of estimating both VaR and CVaR using stochastic approximation (with decreasing steps): we propose a first Robbins-Monro (RM) procedure based on Rockafellar-Uryasev’s identity for the CVaR. Convergence rate of this algorithm to its target satisfies a Gaussian Central Limit Theorem. As a second step, in order to speed up the initial procedure, we propose a recursive and adaptive importance sampling (IS) procedure which induces a significant variance reduction of both VaR and CVaR procedures. This idea, which has been investigated by many authors, follows a new approach introduced in [27]. Finally, to speed up the initialization phase of the IS algorithm, we replace the original confidence level of the VaR by a slowly moving risk level. We prove that the weak convergence rate of the resulting procedure is ruled by a Central Limit Theorem with minimal variance and its efficiency is illustrated on several typical energy portfolios.

Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles

Abstract: We develop a tool to approximate the entries of a large dimensional complex Jacobi ensemble with independent complex Gaussian random variables. Based on this and the author’s earlier work in this direction, we obtain the Tracy–Widom law of the largest singular values of the Jacobi emsemble. Moreover, the circular law, the Marchenko–Pastur law, the central limit theorem, and the laws of large numbers for the spectral norms are also obtained.

Stein's method meets Malliavin calculus: a short survey with new estimates

Abstract: We provide an overview of some recent techniques involving the Malliavin calculus of variations and the so-called ``Stein's method'' for the Gaussian approximations of probability distributions. Special attention is devoted to establishing explicit connections with the classic method of moments: in particular, we use interpolation techniques in order to deduce some new estimates for the moments of random variables belonging to a fixed Wiener chaos. As an illustration, a class of central limit theorems associated with the quadratic variation of a fractional Brownian motion is studied in detail.

Uniform limit theorems for wavelet density estimators

Abstract: Let pn(y)=∑kαkϕ(y−k)+∑l=0jn−1∑kβlk2l/2ψ(2ly−k) be the linear wavelet density estimator, where ϕ, ψ are a father and a mother wavelet (with compact support), αk, βlk are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density p0 on ℝ, and jn∈ℤ, jn↗∞. Several uniform limit theorems are proved: First, the almost sure rate of convergence of sup y∈ℝ|pn(y)−Epn(y)| is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that sup y∈ℝ|pn(y)−p0(y)| attains the optimal almost sure rate of convergence for estimating p0, if jn is suitably chosen. Second, a uniform central limit theorem as well as strong invariance principles for the distribution function of pn, that is, for the stochastic processes $\sqrt{n}(F_{n}^{W}(s)-F(s))=\sqrt{n}\int_{-\infty}^{s}(p_{n}-p_{0})$, s∈ℝ, are proved; and more generally, uniform central limit theorems for the processes $\sqrt{n}\int(p_{n}-p_{0})f$, $f\in\mathcal{F}$, for other Donsker classes $\mathcal{F}$ of interest are considered. As a statistical application, it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by Donoho et al. [Ann. Statist. 24 (1996) 508–539].

Asymptotic normality of plug-in level set estimates.

Abstract: We establish the asymptotic normality of the G-measure of the symmetric difference between the level set and a plug-in-type estimator of it formed by replacing the density in the definition of the level set by a kernel density estimator. Our proof will highlight the efficacy of Poissonization methods in the treatment of large sample theory problems of this kind.

Limit theorems for Markov processes indexed by continuous time Galton--Watson trees

Abstract: We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton--Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time t. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. The latter has the same generator as the Markov process along the branches plus additional jumps, associated with branching events of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time t favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching Levy processes.

Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example

Abstract: Extremization of the Boltzmann-Gibbs (BG) entropy $S_{BG}=-k\int dx\,p(x) \ln p(x)$ under appropriate norm and width constraints yields the Gaussian distribution pG(x) ∝e-βx. Also, the basic solutions of the standard Fokker-Planck (FP) equation (related to the Langevin equation with additive noise), as well as the Central Limit Theorem attractors, are Gaussians. The simplest stochastic model with such features is N ↦∞ independent binary random variables, as first proved by de Moivre and Laplace. What happens for strongly correlated random variables? Such correlations are often present in physical situations as e.g. systems with long range interactions or memory. Frequently q-Gaussians, pq(x) ∝[1-(1-q)βx2]1/(1-q) [p1(x)=pG(x)] become observed. This is typically so if the Langevin equation includes multiplicative noise, or the FP equation to be nonlinear. Scale-invariance, e.g. exchangeable binary stochastic processes, allow a systematical analysis of the relation between correlations and non-Gaussian distributions. In particular, a generalized stochastic model yielding q-Gaussians for all (q ≠ 1) was missing. This is achieved here by using the Laplace-de Finetti representation theorem, which embodies strict scale-invariance of interchangeable random variables. We demonstrate that strict scale invariance together with q-Gaussianity mandates the associated extensive entropy to be BG.

A Central Limit Theorem for the SINR at the LMMSE Estimator Output for Large-Dimensional Signals

Abstract: This paper is devoted to the performance study of the linear minimum mean squared error (LMMSE) estimator for multidimensional signals in the large-dimension regime. Such an estimator is frequently encountered in wireless communications and in array processing, and the signal-to-interference-plus-noise ratio (SINR) at its output is a popular performance index. The SINR can be modeled as a random quadratic form which can be studied with the help of large random matrix theory, if one assumes that the dimension of the received and transmitted signals go to infinity at the same pace. This paper considers the asymptotic behavior of the SINR for a wide class of multidimensional signal models that includes general multiple-antenna as well as spread-spectrum transmission models. The expression of the deterministic approximation of the SINR in the large-dimension regime is recalled and the SINR fluctuations around this deterministic approximation are studied. These fluctuations are shown to converge in distribution to the Gaussian law in the large-dimension regime, and their variance is shown to decrease as the inverse of the signal dimension.

Billiards in a General Domain with Random Reflections

Abstract: We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain \(\fancyscript{D}\subset {\mathbb{R}}^d\) until it hits the boundary and bounces randomly inside, according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord “picked at random” in \(\fancyscript{D}\) , and we study the angle of intersection of the process with a (d − 1)-dimensional manifold contained in \(\fancyscript{D}\) .

Optimality of the auxiliary particle filter

Abstract: In this article we study asymptotic properties of weighted samples produced by the auxiliary particle filter (APF) proposed by Pitt and Shephard [17]. Besides establishing a central limit theorem (CLT) for smoothed particle estimates, we also derive bounds on the Lp error and bias of the same for a finite particle sample size. By examining the recursive formula for the asymptotic variance of the CLT we identify first- stage importance weights for which the increase of asymptotic variance at a single iteration of the algorithm is minimal. In the light of these findings, we discuss and demonstrate on several examples how the APF algorithm can be improved.

On the distribution of the sum of independent uniform random variables

Abstract: Motivated by an application in change point analysis, we derive a closed form for the density function of the sum of n independent, non-identically distributed, uniform random variables.

On the computational complexity of MCMC-based estimators in large samples

Abstract: In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasi- Bayesian estimation carried out using Metropolis random walks. Our anal- ysis is motivated by the Laplace-Bernstein-Von Mises central limit theo- rem, which states that in large samples the posterior or quasi-posterior approaches a normal density. Using this observation, we establish polyno- mial bounds on the computational complexity of general Metropolis ran- dom walks methods in large samples. Our analysis covers cases, where the underlying log-likelihood or extremum criterion function is possibly non- concave, discontinuous, and of increasing dimension. However, the central limit theorem restricts the deviations from continuity and log-concavity of the log-likelihood or extremum criterion function in a very specific manner. Under minimal assumptions for the central limit theorem framework to hold, we show that the Metropolis algorithm is theoretically efficient even for the canonical Gaussian walk which is studied in detail. Specifically, we show that the running time of the algorithm in large samples is bounded in probability by a polynomial in the parameter dimension d, and, in partic- ular, is of stochastic order d2 in the leading cases after the burn-in period. We then give an application to exponential and curved exponential families of increasing dimension.