Showing papers on "Central limit theorem published in 2004"

Feynman-Kac formulae : genealogical and interacting particle systems with applications

Abstract: 1 Introduction- 11 On the Origins of Feynman-Kac and Particle Models- 12 Notation and Conventions- 13 Feynman-Kac Path Models- 131 Path-Space and Marginal Models- 132 Nonlinear Equations- 14 Motivating Examples- 141 Engineering Science- 142 Bayesian Methodology- 143 Particle and Statistical Physics- 144 Biology- 145 Applied Probability and Statistics- 15 Interacting Particle Systems- 151 Discrete Time Models- 152 Continuous Time Models- 16 Sequential Monte Carlo Methodology- 17 Particle Interpretations- 18 A Contents Guide for the Reader- 2 Feynman-Kac Formulae- 21 Introduction- 22 An Introduction to Markov Chains- 221 Canonical Probability Spaces- 222 Path-Space Markov Models- 223 Stopped Markov chains- 224 Examples- 23 Description of the Models- 24 Structural Stability Properties- 241 Path Space and Marginal Models- 242 Change of Reference Probability Measures- 243 Updated and Prediction Flow Models- 25 Distribution Flows Models- 251 Killing Interpretation- 252 Interacting Process Interpretation- 253 McKean Models- 254 Kalman-Bucy filters- 26 Feynman-Kac Models in Random Media- 261 Quenched and Annealed Feynman-Kac Flows- 262 Feynman-Kac Models in Distribution Space- 27 Feynman-Kac Semigroups- 271 Prediction Semigroups- 272 Updated Semigroups- 3 Genealogical and Interacting Particle Models- 31 Introduction- 32 Interacting Particle Interpretations- 33 Particle models with Degenerate Potential- 34 Historical and Genealogical Tree Models- 341 Introduction- 342 A Rigorous Approach and Related Transport Problems- 343 Complete Genealogical Tree Models- 35 Particle Approximation Measures- 351 Some Convergence Results- 352 Regularity Conditions- 4 Stability of Feynman-Kac Semigroups- 41 Introduction- 42 Contraction Properties of Markov Kernels- 421 h-relative Entropy- 422 Lipschitz Contractions- 43 Contraction Properties of Feynman-Kac Semigroups- 431 Functional Entropy Inequalities- 432 Contraction Coefficients- 433 Strong Contraction Estimates- 434 Weak Regularity Properties- 44 Updated Feynman-Kac Models- 45 A Class of Stochastic Semigroups- 5 Invariant Measures and Related Topics- 51 Introduction- 52 Existence and Uniqueness- 53 Invariant Measures and Feynman-Kac Modeling- 54 Feynman-Kac and Metropolis-Hastings Models- 55 Feynman-Kac-Metropolis Models- 551 Introduction- 552 The Genealogical Metropolis Particle Model- 553 Path Space Models and Restricted Markov Chains- 554 Stability Properties- 6 Annealing Properties- 61 Introduction- 62 Feynman-Kac-Metropolis Models- 621 Description of the Model- 622 Regularity Properties- 623 Asymptotic Behavior- 63 Feynman-Kac Trapping Models- 631 Description of the Model- 632 Regularity Properties- 633 Asymptotic Behavior- 634 Large-Deviation Analysis- 635 Concentration Levels- 7 Asymptotic Behavior- 71 Introduction- 72 Some Preliminaries- 721 McKean Interpretations- 722 Vanishing Potentials- 73 Inequalities for Independent Random Variables- 731 Lp and Exponential Inequalities- 732 Empirical Processes- 74 Strong Law of Large Numbers- 741 Extinction Probabilities- 742 Convergence of Empirical Processes- 743 Time-Uniform Estimates- 8 Propagation of Chaos- 81 Introduction- 82 Some Preliminaries- 83 Outline of Results- 84 Weak Propagation of Chaos- 85 Relative Entropy Estimates- 86 A Combinatorial Transport Equation- 87 Asymptotic Properties of Boltzmann-Gibbs Distributions- 88 Feynman-Kac Semigroups- 881 Marginal Models- 882 Path-Space Models- 89 Total Variation Estimates- 9 Central Limit Theorems- 91 Introduction- 92 Some Preliminaries- 93 Some Local Fluctuation Results- 94 Particle Density Profiles- 941 Unnormalized Measures- 942 Normalized Measures- 943 Killing Interpretations and Related Comparisons- 95 A Berry-Esseen Type Theorem- 96 A Donsker Type Theorem- 97 Path-Space Models- 98 Covariance Functions- 10 Large-Deviation Principles- 101 Introduction- 102 Some Preliminary Results- 1021 Topological Properties- 1022 Idempotent Analysis- 1023 Some Regularity Properties- 103 Cramer's Method- 104 Laplace-Varadhan's Integral Techniques- 105 Dawson-Gartner Projective Limits Techniques- 106 Sanov's Theorem- 1061 Introduction- 1062 Topological Preliminaries- 1063 Sanov's Theorem in the r-Topology- 107 Path-Space and Interacting Particle Models- 1071 Proof of Theorem 1011- 1072 Sufficient Conditions- 108 Particle Density Profile Models- 1081 Introduction- 1082 Strong Large-Deviation Principles- 11 Feynman-Kac and Interacting Particle Recipes- 111 Introduction- 112 Interacting Metropolis Models- 1121 Introduction- 1122 Feynman-Kac-Metropolis and Particle Models- 1123 Interacting Metropolis and Gibbs Samplers- 113 An Overview of some General Principles- 114 Descendant and Ancestral Genealogies- 115 Conditional Explorations- 116 State-Space Enlargements and Path-Particle Models- 117 Conditional Excursion Particle Models- 118 Branching Selection Variants- 1181 Introduction- 1182 Description of the Models- 1183 Some Branching Selection Rules- 1184 Some L2-mean Error Estimates- 1185 Long Time Behavior- 1186 Conditional Branching Models- 119 Exercises- 12 Applications- 121 Introduction- 122 Random Excursion Models- 1221 Introduction- 1222 Dirichlet Problems with Boundary Conditions- 1223 Multilevel Feynman-Kac Formulae- 1224 Dirichlet Problems with Hard Boundary Conditions- 1225 Rare Event Analysis- 1226 Asymptotic Particle Analysis of Rare Events- 1227 Fluctuation Results and Some Comparisons- 1228 Exercises- 123 Change of Reference Measures- 1231 Introduction- 1232 Importance Sampling- 1233 Sequential Analysis of Probability Ratio Tests- 1234 A Multisplitting Particle Approach- 1235 Exercises- 124 Spectral Analysis of Feynman-Kac-Schrodinger Semigroups- 1241 Lyapunov Exponents and Spectral Radii- 1242 Feynman-Kac Asymptotic Models- 1243 Particle Lyapunov Exponents- 1244 Hard, Soft and Repulsive Obstacles- 1245 Related Spectral Quantities- 1246 Exercises- 125 Directed Polymers Simulation- 1251 Feynman-Kac and Boltzmann-Gibbs Models- 1252 Evolutionary Particle Simulation Methods- 1253 Repulsive Interaction and Self-Avoiding Markov Chains- 1254 Attractive Interaction and Reinforced Markov Chains- 1255 Particle Polymerization Techniques- 1256 Exercises- 126 Filtering/Smoothing and Path estimation- 1261 Introduction- 1262 Motivating Examples- 1263 Feynman-Kac Representations- 1264 Stability Properties of the Filtering Equations- 1265 Asymptotic Properties of Log-likelihood Functions- 1266 Particle Approximation Measures- 1267 A Partially Linear/Gaussian Filtering Model- 1268 Exercises- References

On the Markov chain central limit theorem

Abstract: The goal of this expository paper is to describe conditions which guarantee a central limit theorem for functionals of general state space Markov chains. This is done with a view towards Markov chain Monte Carlo settings and hence the focus is on the connections between drift and mixing conditions and their implications. In particular, we consider three commonly cited central limit theorems and discuss their relationship to classical results for mixing processes. Several motivating examples are given which range from toy one-dimensional settings to complicated settings encountered in Markov chain Monte Carlo.

Information Theory And The Central Limit Theorem

Abstract: This book provides a comprehensive description of a new method of proving the central limit theorem, through the use of apparently unrelated results from information theory. It gives a basic introduction to the concepts of entropy and Fisher information, and collects together standard results concerning their behaviour. It brings together results from a number of research papers as well as unpublished material, showing how the techniques can give a unified view of limit theorems.

Quenched invariance principles for walks on clusters of percolation or among random conductances

Abstract: In this work we principally study random walk on the supercritical infinite cluster for bond percolation on ℤd. We prove a quenched functional central limit theorem for the walk when d≥4. We also prove a similar result for random walk among i.i.d. random conductances along nearest neighbor edges of ℤd, when d≥1.

Limit theorems for iterated random functions

Abstract: We study geometric moment contracting properties of nonlinear time series that are expressed in terms of iterated random functions. Under a Dini-continuity condition, a central limit theorem for additive functionals of such systems is established. The empirical processes of sample paths are shown to converge to Gaussian processes in the Skorokhod space. An exponential inequality is established. We present a bound for joint cumulants, which ensures the applicability of several asymptotic results in spectral analysis of time series. Our results provide a vehicle for statistical inferences for fractals and many nonlinear time series models.

Solution of Shannon's problem on the monotonicity of entropy

Abstract: It is shown that if X1, X2, . . . are independent and identically distributed square-integrable random variables then the entropy of the normalized sum Ent (X1+ · · · + Xn over √n) is an increasing function of n. This resolves an old problem which goes back to [6, 7, 5]. The result also has a version for non-identically distributed random variables or random vectors.

Fisher Information inequalities and the Central Limit Theorem

Abstract: We give conditions for an O(1/n) rate of convergence of Fisher information and relative entropy in the Central Limit Theorem. We use the theory of projections in L 2 spaces and Poincare inequalities, to provide a better understanding of the decrease in Fisher information implied by results of Barron and Brown. We show that if the standardized Fisher information ever becomes finite then it converges to zero.

Sharp polynomial estimates for the decay of correlations

Abstract: We generalize a method developed by Sarig to obtain polynomial lower bounds for correlation functions for maps with a countable Markov partition. A consequence is that LS Young’s estimates on towers are always optimal. Moreover, we show that, for functions with zero average, the decay rate is better, gaining a factor 1/n. This implies a Central Limit Theorem in contexts where it was not expected, e.g.,x+Cx 1+α with 1/2⩽α<1. The method is based on a general result on renewal sequences of operators, and gives an asymptotic estimate up to any precision of such operators.

Central limit theorem and stable laws for intermittent maps

Abstract: In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form x+x 1+α , for α∈(0, 1). In particular, for α>1/2, we show that the Birkhoff sums of a Holder observable f converge to a normal law or a stable law, depending on whether f(0)=0 or f(0)≠0. The proof uses spectral techniques introduced by Sarig, and Wiener’s Lemma in non-commutative Banach algebras.

SEQUENTIAL CHANGE-POINT DETECTION IN GARCH(p,q) MODELS

Abstract: We suggest a sequential monitoring scheme to detect changes in the parameters of a GARCH(p,q) sequence. The procedure is based on quasi-likelihood scores and does not use model residuals. Unlike for linear regression models, the squared residuals of nonlinear time series models such as generalized autoregressive conditional heteroskedasticity (GARCH) do not satisfy a functional central limit theorem with a Wiener process as a limit, so its boundary crossing probabilities cannot be used. Our procedure nevertheless has an asymptotically controlled size, and, moreover, the conditions on the boundary function are very simple; it can be chosen as a constant. We establish the asymptotic properties of our monitoring scheme under both the null of no change in parameters and the alternative of a change in parameters and investigate its finite-sample behavior by means of a small simulation study.This research was partially supported by NSF grant INT-0223262 and NATO grant PST.CLG.977607. The work of the first author was supported by the Hungarian National Foundation for Scientific Research, grants T 29621, 37886; the work of the second author was supported by NSERC Canada.

Testing Granger Causality in Heterogenous Panel Data Models with Fixed Coefficients

Abstract: This paper proposes a simple test of Granger (1969) non causality hypothesis in heterogeneous panel data models with fixed coefficients. It proposes a statistic of test based on averaging standard individual Wald statistics of Granger non causality tests. First, this statistic is shown to converge sequentially to a standard normal distribution with T tends to infinity, followed by N. Second, for a fixed T sample the semi-asymptotic distribution of the average statistic is characterized. In this case, individual Wald statistics do not have a standard distribution. However, under very general setting, we prove that individual Wald statistics are independently distributed with finite second order moments as soon as T> 5+2 K, where K denotes the number of linear restrictions. For a fixed T sample, the Lyapunov central limit theorem is then sufficient to get the semi asymptotic distribution when N tends to infinity. The two first moments of this normal distribution correspond to empirical mean of the corresponding theoretical moments of the individual Wald statistics. The issue is then to propose an evaluation of the two first moments of standard Wald statistics for small T sample. In this paper we propose a general approximation based on the exact moments of the ratio of quadratic forms in normal variables derived from the Magnus (1986) theorem. For a fixed T sample, we propose simple approximations of the mean and the variance of the Wald statistic. Monte Carlo experiments show that these formulas provide an excellent approximation. Given these approximations, we propose an approximated standardized average Wald statistic to test the HNC hypothesis in short T sample. Finally, approximated critical values are proposed for finite N and T sample and compared to simulated critical values in some experiments.

Statistical limit theorems for suspension flows

Abstract: In dynamical systems theory, a standard method for passing from discrete time to continuous time is to construct the suspension flow under a roof function. In this paper, we give conditions under which statistical laws, such as the central limit theorem and almost sure invariance principle, for the underlying discrete time system are inherited by the suspension flow. As a consequence, we give a simpler proof of the results of Ratner (1973) and recover the results of Denker and Philipp (1984) for Axiom A flows. Morcover, we obtain several new results for nonuniformly and partially hyperbolic flows, including frame flows on negatively curved manifolds satisfying a pinching condition.

Martingale approximations for sums of stationary processes

Abstract: Approximations to sums of stationary and ergodic sequences by martingales are investigated. Necessary and sufficient conditions for such sums to be asymptotically normal conditionally given the past up to time 0 are obtained. It is first shown that a martingale approximation is necessary for such normality and then that the sums are asymptotically normal if and only if the approximating martingales satisfy a Lindeberg-Feller condition. Using the explicit construction of the approximating martingales, a central limit theorem is derived for the sample means of linear processes. The conditions are not sufficient for the functional version of the central limit theorem. This is shown by an example, and a slightly stronger sufficient condition is given.

Ruelle's linear response formula, ensemble adjoint schemes and Lévy flights

Abstract: A traditional subject in statistical physics is the linear response of a molecular dynamical system to changes in an external forcing agency, e.g. the Ohmic response of an electrical conductor to an applied electric field. For molecular systems the linear response matrices, such as the electrical conductivity, can be represented by Green–Kubo formulae as improper time-integrals of 2-time correlation functions in the system. Recently, Ruelle has extended the Green–Kubo formalism to describe the statistical, steady-state response of a 'sufficiently chaotic' nonlinear dynamical system to changes in its parameters. This formalism potentially has a number of important applications. For instance, in studies of global warming one wants to calculate the response of climate-mean temperature to a change in the atmospheric concentration of greenhouse gases. In general, a climate sensitivity is defined as the linear response of a long-time average to changes in external forces. We show that Ruelle's linear response formula can be computed by an ensemble adjoint technique and that this algorithm is equivalent to a more standard ensemble adjoint method proposed by Lea, Allen and Haine to calculate climate sensitivities.In a numerical implementation for the 3-variable, chaotic Lorenz model it is shown that the two methods perform very similarly. However, because of a power-law tail in the histogram of adjoint gradients their sum over ensemble members becomes a Levy flight, and the central limit theorem breaks down. The law of large numbers still holds and the ensemble-average converges to the desired sensitivity, but only very slowly, as the number of samples is increased. We discuss the implications of this example more generally for ensemble adjoint techniques and for the important practical issue of calculating climate sensitivities.

Rate of convergence in probability to the Marchenko-Pastur law

Abstract: It is shown that the Kolmogorov distance between the spectral distribution function of a random covariance matrix (1/p)XXT, where X is an n×p matrix with independent entries and the distribution function of the Marchenko-Pastur law is of order O(n-1/2) in probability. The bound is explicit and requires that the twelfth moment of the entries of the matrix is uniformly bounded and that p/n is separated from 1.

The High Temperature Region of the Viana-Bray Diluted Spin Glass Model*

Abstract: In this paper, we study the high temperature or low connectivity phase of the Viana–Bray model in the absence of magnetic field. This is a diluted version of the well known Sherrington–Kirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infinite volume free energy and a central limit theorem for the suitably rescaled fluctuations of the multi-overlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a non-Gaussian random variable, whose variance diverges at the boundary of the replica-symmetric region. The connection with the fully connected Sherrington– Kirkpatrick model is discussed.

The Bernoulli sieve

Abstract: The Bernoulli sieve is a recursive construction of a random composition (ordered partition) of an integer n. This composition can be induced by sampling from a random discrete distribution which has frequencies equal to the sizes of components of a stick-breaking interval partition of (0, 1). We exploit the Markov property of the composition and its renewal representation to study the number of its parts. We derive asymptotics of the moments and prove a central limit theorem.

On the rate of convergence in the entropic central limit theorem

Abstract: We study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition.

M-estimation of linear models with dependent errors

Abstract: We study asymptotic properties of $M$-estimates of regression parameters in linear models in which errors are dependent. Weak and strong Bahadur representations of the $M$-estimates are derived and a central limit theorem is established. The results are applied to linear models with errors being short-range dependent linear processes, heavy-tailed linear processes and some widely used nonlinear time series.

Periodic Homogenization for Hypoelliptic Diffusions

Abstract: We study the long time behavior of an Ornstein–Uhlenbeck process under the influence of a periodic drift. We prove that, under the standard diffusive rescaling, the law of the particle position converges weakly to the law of a Brownian motion whose covariance can be expressed in terms of the solution of a Poisson equation. We also derive upper bounds on the convergence rate in several metrics.

Tail Asymptotics for the Supremum of a Random Walk when the Mean Is not Finite

Abstract: We consider the sums Sneξ1+⋅⋅⋅+ξn of independent identically distributed random variables. We do not assume that the ξ's have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability P{M>x} as x→∞, provided that Mesup {Sn,ng1} is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite- and the finite-mean cases. In particular, we show that the subexponentiality of distribution F does not imply the subexponentiality of its integrated tail distribution FI.

Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive

Abstract: We consider a general model of discrete-time random walk X t on the lattice ν, ν = 1,..., in a random environment ξ={ξ(t,x):(t,x)∈ ν+1} with i.i.d. components ξ(t,x). Previous results on the a.s. validity of the Central Limit Theorem for the quenched model required a small stochasticity condition. In this paper we show that the result holds provided only that an obvious non-degeneracy condition is met. The proof is based on the analysis of a suitable generating function, which allows to estimate L 2 norms by contour integrals.

An approximate capacity distribution for MIMO systems

Abstract: In this letter, we derive the exact variance of the capacity of a multiple-input multiple-output (MIMO) system. This enables an investigation of the accuracy of a Gaussian approximation to the capacity foreshadowed by various central limit theorems. We confirm recent results which state that the capacity variance appears to converge to a limit independent of absolute antenna numbers, but dependent on the ratio of the numbers of receive to transmit antennas. The Gaussian approximation itself is surprisingly good, even in the worst cases giving satisfactory results.

Stein’s method and Plancherel measure of the symmetric group

Abstract: We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group.

Geometric Ergodicity of van Dyk and Meng's Algorithm for the Multivariate Student's t Model

Abstract: Let π denote the posterior distribution that results when a random sample of size n from a d-dimensional location-scale Student's t distribution (with ν degrees of freedom) is combined with the standard noninformative prior. van Dyk and Meng developed an efficient Markov chain Monte Carlo (MCMC) algorithm for sampling from π and provided considerable empirical evidence to suggest that their algorithm converges to stationarity much faster than the standard data augmentation algorithm. In addition to its practical importance, this algorithm is interesting from a theoretical standpoint because it is based upon a Markov chain that is not positive recurrent. In this article, we formally analyze the relevant sub-Markov chain underlying van Dyk and Meng's algorithm. In particular, we establish drift and minorization conditions that show that, for many (d, ν, n) triples, the sub-Markov chain is geometrically ergodic. This is the first general, rigorous analysis of an MCMC algorithm based upon a nonpositive recurr...

Central limit theorems for iterated random Lipschitz mappings

Abstract: Let M be a noncompact metric space in which every closed ball is compact, and let G be a semigroup of Lipschitz mappings of M. Denote by (Yn)n≥1 a sequence of independent G-valued, identically distributed random variables (r.v.’s), and by Z an M-valued r.v. which is independent of the r.v. Yn, n≥1. We consider the Markov chain (Zn)n≥0 with state space M which is defined recursively by Z0=Z and Zn+1=Yn+1Zn for n≥0. Let ξ be a real-valued function on G×M. The aim of this paper is to prove central limit theorems for the sequence of r.v.’s (ξ(Yn,Zn−1))n≥1. The main hypothesis is a condition of contraction in the mean for the action on M of the mappings Yn; we use a spectral method based on a quasi-compactness property of the transition probability of the chain mentioned above, and on a special perturbation theorem.

On the contraction method with degenerate limit equation

Abstract: A class of random recursive sequences (Yn) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form $X\stackrel {\mathcal{L}}{=}X$. For nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the “unique” solution of the limit equation. In this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation. In particular, we establish some new tools and a general convergence scheme, which transfers information on mean and variance into a central limit law (with normal limit). We also obtain a convergence rate result. For the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables.

Stein’s method and non-reversible Markov chains

Abstract: Let W be either the number of descents or inversions of a permutation. Stein's method is applied to show that W satisfies a central limit theorem with error rate n^(-1/2). The construction of an exchangeable pair (W,W') used in Stein's method is non-trivial and uses a non-reversible Markov chain.