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Showing papers on "Central limit theorem published in 1992"


Journal ArticleDOI
TL;DR: In this paper, the authors connect various topological and probabilistic forms of stability for discrete-time Markov chains, including tightness and Harris recurrence, and show that these concepts are largely equivalent for a major class of chains (chains with continuous components), or if the state space has a sufficiently rich class of appropriate sets.
Abstract: In this paper we connect various topological and probabilistic forms of stability for discrete-time Markov chains. These include tightness on the one hand and Harris recurrence and ergodicity on the other. We show that these concepts of stability are largely equivalent for a major class of chains (chains with continuous components), or if the state space has a sufficiently rich class of appropriate sets ('petite sets'). We use a discrete formulation of Dynkin's formula to establish unified criteria for these stability concepts, through bounding of moments of first entrance times to petite sets. This gives a generalization of Lyapunov-Foster criteria for the various stability conditions to hold. Under these criteria, ergodic theorems are shown to be valid even in the non-irreducible case. These results allow a more general test function approach for determining rates of convergence of the underlying distributions of a Markov chain, and provide strong mixing results and new versions of the central limit theorem and the law of the iterated logarithm.

309 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the almost sure convergence of sums of negatively dependent random variables, in particular, the classical strong law of large numbers for independent distributed random variables is generalized to the case of pairwise negative quadrant dependent Random variables.

304 citations


Journal ArticleDOI
TL;DR: In this paper, a direct proof of Voiculescu's addition theorem for real-valued random variables using resolvents of self-adjoint operators is given.

224 citations


Journal ArticleDOI
TL;DR: In this paper, the authors extend Stein's work to prove a central limit theorem for the variance reduction of LHS integrals, showing that the extent of variance reduction depends on the extent to which the integrand is additive.
Abstract: SUMMARY Latin hypercube sampling (LHS) is a technique for Monte Carlo integration, due to McKay, Conover and Beckman. M. Stein proved that LHS integrals have smaller variance than independent and identically distributed Monte Carlo integration, the extent of the variance reduction depending on the extent to which the integrand is additive. We extend Stein's work to prove a central limit theorem. Variance estimation methods for nonparametric regression can be adapted to provide N'12-consistent estimates of the asymptotic variance in LHS. Moreover the skewness can be estimated at this rate. The variance reduction may be explained in terms of certain control variates that cannot be directly measured. We also show how to combine control variates with LHS. Finally we show how these results lead to a frequentist approach to computer experimentation.

208 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied unimodal interval mapsT with negative Schwarzian derivative satisfying the Collet-Eckmann condition |DTn(Tc)|≧Kλcn for some constantsK>0 and λc>1 (c is the critical point ofT).
Abstract: We study unimodal interval mapsT with negative Schwarzian derivative satisfying the Collet-Eckmann condition |DTn(Tc)|≧Kλcn for some constantsK>0 and λc>1 (c is the critical point ofT). We prove exponential mixing properties of the unique invariant probability density ofT, describe the long term behaviour of typical (in the sense of Lebesgue measure) trajectories by Central Limit and Large Deviations Theorems for partial sum processes of the form\(S_n = \Sigma _{i = 0}^{n - 1} f(T^i x)\), and study the distribution of “typical” periodic orbits, also in the sense of a Central Limit Theorem and a Large Deviations Theorem.

175 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider the asymptotic properties of the two-slice method, obtaining simple conditions for convergence and asyptotic normality for sums of conditionally independent random variables.
Abstract: Sliced inverse regression [Li (1989), (1991) and Duan and Li (1991)] is a nonparametric method for achieving dimension reduction in regression problems. It is widely applicable, extremely easy to implement on a computer and requires no nonparametric smoothing devices such as kernel regression. If $Y$ is the response and $X \in \mathbf{R}^p$ is the predictor, in order to implement sliced inverse regression, one requires an estimate of $\Lambda = E\{\operatorname{cov}(X\mid Y)\} = \operatorname{cov}(X) - \operatorname{cov}\{E(X\mid Y)\}$. The inverse regression of $X$ on $Y$ is clearly seen in $\Lambda$. One such estimate is Li's (1991) two-slice estimate, defined as follows: The data are sorted on $Y$, then grouped into sets of size 2, the covariance of $X$ is estimated within each group and these estimates are averaged. In this paper, we consider the asymptotic properties of the two-slice method, obtaining simple conditions for $n^{1/2}$-convergence and asymptotic normality. A key step in the proof of asymptotic normality is a central limit theorem for sums of conditionally independent random variables. We also study the asymptotic distribution of Greenwood's statistics in nonuniform cases.

165 citations


Journal ArticleDOI
TL;DR: In this paper, the authors give a simple upper bound on the total variation distance between a random permutation and an independent Poisson process and show that this distance decays to zero superexponentially fast as a function of n/b \rightarrow \infty.
Abstract: The total variation distance between the process which counts cycles of size $1,2,\ldots, b$ of a random permutation of $n$ objects and a process $(Z_1,Z_2,\ldots, Z_b)$ of independent Poisson random variables with $\mathbb{E}Z_i = 1/i$ converges to 0 if and only if $b/n \rightarrow 0$. This Poisson approximation can be used to give simple proofs of limit theorems and bounds for a wide variety of functionals of random permutations. These limit theorems include the Erdos-Turan theorem for the asymptotic normality of the log of the order of a random permutation, and the DeLaurentis-Pittel functional central limit theorem for the cycle sizes. We give a simple explicit upper bound on the total variation distance to show that this distance decays to zero superexponentially fast as a function of $n/b \rightarrow \infty$. A similar result holds for derangements and, more generally, for permutations conditioned to have given numbers of cycles of various sizes. Comparison results are included to show that in approximating the cycle structure by an independent Poisson process the main discrepancy arises from independence rather than from Poisson marginals.

153 citations


Book
01 Jan 1992
TL;DR: In this paper, a model of Spatial Distribution is used to estimate the least square estimation of random variables and their transformations, and the results are used to predict the probability of an event.
Abstract: 1 Uncertainty, Intuition, and Expectation.- 1 Ideas and Examples.- 2 The Empirical Basis.- 3 Averages over a Finite Population.- 4 Repeated Sampling: Expectation.- 5 More on Sample Spaces and Variables.- 6 Ideal and Actual Experiments: Observables.- 2 Expectation.- 1 Random Variables.- 2 Axioms for the Expectation Operator.- 3 Events: Probability.- 4 Some Examples of an Expectation.- 5 Moments.- 6 Applications: Optimization Problems.- 7 Equiprobable Outcomes: Sample Surveys.- 8 Applications: Least Square Estimation of Random Variables.- 9 Some Implications of the Axioms.- 3 Probability.- 1 Events, Sets and Indicators.- 2 Probability Measure.- 3 Expectation as a Probability Integral.- 4 Some History.- 5 Subjective Probability.- 4 Some Basic Models.- 1 A Model of Spatial Distribution.- 2 The Multinomial, Binomial, Poisson and Geometric Distributions.- 3 Independence.- 4 Probability Generating Functions.- 5 The St. Petersburg Paradox.- 6 Matching, and Other Combinatorial Problems.- 7 Conditioning.- 8 Variables on the Continuum: The Exponential and Gamma Distributions.- 5 Conditioning.- 1 Conditional Expectation.- 2 Conditional Probability.- 3 A Conditional Expectation as a Random Variable.- 4 Conditioning on a ? Field.- 5 Independence.- 6 Statistical Decision Theory.- 7 Information Transmission.- 8 Acceptance Sampling.- 6 Applications of the Independence Concept.- 1 Renewal Processes.- 2 Recurrent Events: Regeneration Points.- 3 A Result in Statistical Mechanics: The Gibbs Distribution.- 4 Branching Processes.- 7 The Two Basic Limit Theorems.- 1 Convergence in Distribution (Weak Convergence).- 2 Properties of the Characteristic Function.- 3 The Law of Large Numbers.- 4 Normal Convergence (the Central Limit Theorem).- 5 The Normal Distribution.- 6 The Law of Large Numbers and the Evaluation of Channel Capacity.- 8 Continuous Random Variables and Their Transformations.- 1 Distributions with a Density.- 2 Functions of Random Variables.- 3 Conditional Densities.- 9 Markov Processes in Discrete Time.- 1 Stochastic Processes and the Markov Property.- 2 The Case of a Discrete State Space: The Kolmogorov Equations.- 3 Some Examples: Ruin, Survival and Runs.- 4 Birth and Death Processes: Detailed Balance.- 5 Some Examples We Should Like to Defer.- 6 Random Walks, Random Stopping and Ruin.- 7 Auguries of Martingales.- 8 Recurrence and Equilibrium.- 9 Recurrence and Dimension.- 10 Markov Processes in Continuous Time.- 1 The Markov Property in Continuous Time.- 2 The Case of a Discrete State Space.- 3 The Poisson Process.- 4 Birth and Death Processes.- 5 Processes on Nondiscrete State Spaces.- 6 The Filing Problem.- 7 Some Continuous-Time Martingales.- 8 Stationarity and Reversibility.- 9 The Ehrenfest Model.- 10 Processes of Independent Increments.- 11 Brownian Motion: Diffusion Processes.- 12 First Passage and Recurrence for Brownian Motion.- 11 Action Optimisation Dynamic Programming.- 1 Action Optimisation.- 2 Optimisation over Time: the Dynamic Programming Equation.- 3 State Structure.- 4 Optimal Control Under LQG Assumptions.- 5 Minimal-Length Coding.- 6 Discounting.- 7 Continuous-Time Versions and Infinite-Horizon Limits.- 8 Policy Improvement.- 12 Optimal Resource Allocation.- 1 Portfolio Selection in Discrete Time.- 2 Portfolio Selection in Continuous Time.- 3 Multi-Armed Bandits and the Gittins Index.- 4 Open Processes.- 5 Tax Problems.- 13 Finance: 'Risk-Free' Trading and Option Pricing.- 1 Options and Hedging Strategies.- 2 Optimal Targeting of the Contract.- 3 An Example.- 4 A Continuous-Time Model.- 5 How Should it Be Done?.- 14 Second-Order Theory.- 1 Back to L2.- 2 Linear Least Square Approximation.- 3 Projection: Innovation.- 4 The Gauss-Markov Theorem.- 5 The Convergence of Linear Least Square Estimates.- 6 Direct and Mutual Mean Square Convergence.- 7 Conditional Expectations as Least Square Estimates: Martingale Convergence.- 15 Consistency and Extension: The Finite-Dimensional Case.- 1 The Issues.- 2 Convex Sets.- 3 The Consistency Condition for Expectation Values.- 4 The Extension of Expectation Values.- 5 Examples of Extension.- 6 Dependence Information: Chernoff Bounds.- 16 Stochastic Convergence.- 1 The Characterization of Convergence.- 2 Types of Convergence.- 3 Some Consequences.- 4 Convergence in rth Mean.- 17 Martingales.- 1 The Martingale Property.- 2 Kolmogorov's Inequality: the Law of Large Numbers.- 3 Martingale Convergence: Applications.- 4 The Optional Stopping Theorem.- 5 Examples of Stopped Martingales.- 18 Large-Deviation Theory.- 1 The Large-Deviation Property.- 2 Some Preliminaries.- 3 Cramer's Theorem.- 4 Some Special Cases.- 5 Circuit-Switched Networks and Boltzmarm Statistics.- 6 Multi-Class Traffic and Effective Bandwidth.- 7 Birth and Death Processes.- 19 Extension: Examples of the Infinite-Dimensional Case.- 1 Generalities on the Infinite-Dimensional Case.- 2 Fields and ?-Fields of Events.- 3 Extension on a Linear Lattice.- 4 Integrable Functions of a Scalar Random Variable.- 5 Expectations Derivable from the Characteristic Function: Weak Convergence324.- 20 Quantum Mechanics.- 1 The Static Case.- 2 The Dynamic Case.- References.

150 citations


Journal ArticleDOI
TL;DR: In this article, the authors established general conditions for the asymptotic validity of sequential stopping rules to achieve fixed-volume confidence sets for simulation estimators of vector-valued parameters.
Abstract: : We establish general conditions for the asymptotic validity of sequential stopping rules to achieve fixed-volume confidence sets for simulation estimators of vector-valued parameters. The asymptotic validity occurs as the prescribed volume of the confidences set approaches zero. There are two requirements: a functional central limit theorem for the estimation process and strong consistency (with-probability-one convergence) for the variance or scaling matrix estimator. Applications are given for: sample means of i.i.d. random variables and random vectors, nonlinear functions of such sample means, jackknifing, Kiefer-Wolfowitz and Robbins-Monro stochastic approximation, and both regenerative and non-regenerative steady-state simulation. Keywords: Stochastic simulation, Variance estimators.

137 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of estimating the transfer function of a linear system, together with the spectral density of an additive disturbance, and showed that the estimates are strongly consistent and asymptotically normal.
Abstract: The problem of estimating the transfer function of a linear system, together with the spectral density of an additive disturbance, is considered. The set of models used consists of linear rational transfer functions and the spectral densities are estimated from a finite-order autoregressive disturbance description. The true system and disturbance spectrum are, however, not necessarily of finite order. We investigate the properties of the estimates obtained as the number of observations tends to ∞ at the same time as the model order employed tends to ∞ . It is shown that the estimates are strongly consistent and asymptotically normal, and an expression for the asymptotic variances is also given. The variance of the transfer function estimate at a certain frequency is related to the signal/noise ratio at that frequency and the model orders used, as well as the number of observations. The variance of the noise spectral estimate relates in a similar way to the squared value of the true spectrum.

124 citations


Journal ArticleDOI
TL;DR: For weakly stationary random fields, conditions on coefficients of "linear dependence" are given which are, respectively, sufficient and sufficient for the existence of a continuous spectral density as discussed by the authors.
Abstract: For weakly stationary random fields, conditions on coefficients of “linear dependence” are given which are, respectively, sufficient for the existence of a continuous spectral density, and necessary and sufficient for the existence of a continuous positive spectral density. For strictly stationary random fields, central limit theorems are proved under the corresponding “unrestricted ϱ-mixing” condition and just finite or “barely infinite” second moments. No mixing rate is assumed.

Journal ArticleDOI
Allan Gut1
TL;DR: In this article, the authors extend and generalize some recent results due to Hu, Moricz and Taylor concerning complete convergence, in the sense of Hsu and Robbins, of the sequence of rowwise arithmetic means.
Abstract: Let {(X nk , 1≤k≤n),n≥1}, be an array of rowwise independent random variables. We extend and generalize some recent results due to Hu, Moricz and Taylor concerning complete convergence, in the sense of Hsu and Robbins, of the sequence of rowwise arithmetic means.

Journal ArticleDOI
TL;DR: Fluctuation of the mean field is studied for a network of chaotic elements with the use of globally coupled maps as the size of the chaotic elements increases as discussed by the authors, where the remaining variance is roughly proportional to 1Nc.

Journal ArticleDOI
TL;DR: In this article, the generalized quantile process (GQP) was introduced for random vectors taking values in R = √ n, where n is the number of points in a set.
Abstract: For random vectors taking values in $\mathbb{R}^d$ we introduce a notion of multivariate quantiles defined in terms of a class of sets and study an associated process which we call the generalized quantile process. This process specializes to the well known univariate quantile process. We obtain functional central limit theorems for our generalized quantile process and show that both Gaussian and non-Gaussian limiting processes can arise. A number of interesting example are included.

Journal ArticleDOI
TL;DR: The power ofrete functional limitTheorems to provide elementary proofs of a variety of new and old limit theorems, including results previously proved by complicated analytical methods are demonstrated.
Abstract: Discrete functional limit theorems, which give independent process approximations for the joint distribution of the component structure of combinatorial objects such as permutations and mappings, have recently become available. In this article, we demonstrate the power of these theorems to provide elementary proofs of a variety of new and old limit theorems, including results previously proved by complicated analytical methods. Among the examples we treat are Brownian motion limit theorems for the cycle counts of a random permutation or the component counts of a random mapping, a Poisson limit law for the core of a random mapping, a generalization of the Erdos-Turin Law for the log-order of a random permutation and the smallest component size of a random permutation, approximations to the joint laws of the smallest cycle sizes of a random mapping, and a limit distribution for the difference between the total number of cycles and the number of distinct cycle sizes in a random permutation. @ 1992 John Wiley & Sons, Inc.

Journal ArticleDOI
TL;DR: In this paper, a generalized Fejer theorem was used to show that the periodogram computed from a stationary increment process has non-integrable time-invariant spectra.
Abstract: By means of a generalized Fejer theorem, certain stationary increment random functions and random fields are shown to possess nonintegrable time invariant spectra. The periodogram computed from a stationary increment process is shown to have the same sort of asymptotic statistical properties (e.g., central limit theorem) as the periodogram of a stationary process does. These results throw light on the paradoxical nature of Flicker noise and related phenomena that occur with random fields.

Journal ArticleDOI
TL;DR: In this article, a method of obtaining an M-estimator in a linear model when the responses are subject to right censoring is proposed, and the central limit theorem for the estimator using squared error loss, i.e. least squares, is derived using counting process martingale techniques.
Abstract: SUMMARY We propose a method of obtaining an M-estimator in a linear model when the responses are subject to right censoring. The central limit theorem for the estimator using squared error loss, i.e. least squares, is derived using counting process martingale techniques. The estimation method is applied to the Stanford heart transplant data for illustration.

Journal ArticleDOI
TL;DR: In this article, the authors discussed the failure of the Gaussian approximation to the distribution of the maximum likelihood estimator in one-parameter families for finite sample sizes and derived sufficient conditions on the sample size for Fisher's result to break down.
Abstract: The author discusses the failure of the Gaussian approximation to the distribution of the maximum-likelihood estimator in one-parameter families for finite sample sizes. Fisher (1925) has shown that this approximation is valid when an asymptotically large sample of data points is used. He did this by treating the likelihood equation (i.e. the equation obtained by setting the derivative of the likelihood function with respect to the parameter to zero) statistically and finding its solution as the sample size n is taken to infinity. The statistical treatment of the likelihood equation is extended to include corrections for finite sample sizes. The O(1/n) corrections to Fisher's asymptotic Gaussian result are calculated with corrections to the central limit theorem, and are used to derive sufficient conditions on the sample size for Fisher's result to break down. Such conditions are useful for the design of experiments. The procedure developed here can be extended to the maximum-likelihood estimation of several parameters in multivariate distributions.

Journal ArticleDOI
TL;DR: In this article, a central limit theorem for dependent stochastic processes is proved for the case of martingale differences due to McLeish and suitably defined Bernstein blocks satisfy the required conditions.
Abstract: A central limit theorem is proved for dependent stochastic processes. Global heterogeneity of the distribution of the terms is permitted, including asymptotically unbounded moments. The approach is to adapt a CLT for martingale differences due to McLeish and show that suitably defined Bernstein blocks satisfy the required conditions.


Journal ArticleDOI
TL;DR: In this article, the distribution of the naive bootstrap of the pivot √n(Pn-P) is shown to appropriate that of the pivoting as n↦∞.

Journal ArticleDOI
TL;DR: In this article, the authors developed a one-step triangular array asymptotic expansion for continuous martingales, which are asymPTically normal and satisfy a law of large numbers and a central limit type condition.
Abstract: The paper develops a one-step triangular array asymptotic expansion for continuous martingales which are asymptotically normal. Mixing conditions are not required, but the quadratic variations of the martingales must satisfy a law of large numbers and a central limit type condition. From this result we derive expansions for the distributions of estimators in asymptotically ergodic differential equation models, and also for the bootstrapping estimators of these distributions.

Journal ArticleDOI
TL;DR: In this paper, Bender and Richmond's central and local limit theorems are extended to cover a wider class of generating functions, which will cover the above-mentioned combinatorial structures.

Book ChapterDOI
01 Jan 1992
TL;DR: In this paper, the convergence of moments of the Marcinkiewicz law of large numbers for U-statistics is considered. But the convergence is not considered for the central limit theorem.
Abstract: Laws of large numbers for U-statistics which reduce to the (sufficiencypart of the) Marcinkiewicz law of large numbers if m = 1 are proved. Convergence of moments is also considered both for the laws of large numbers and for the central limit theorem.

Journal Article
TL;DR: In this article, it is shown that a suitably normalized version of the estimated conditional survival function can be represented as a sum of a mean zero local square integrable martingale and a predictable process.
Abstract: To estimate the conditional survival function from a sample of possibly censored survival times and an associated covariate, a kernel estimate is considered. It is shown that in analogy to the uncensored case, a suitably normalized version of the estimated conditional survival function can be represented as a sum of a mean zero local square integrable martingale and a predictable process. The first of these terms accounts for the variance of the estimate whereas the second, giving the history, is deterministic and accounts for the bias. Using Reboliedo's central limit theorem along with tightness criteria of Bickel & Wichura, it is shown that the standardized estimate, considered as a function of time and the unknown bandwidth, converges weakly to a Gaussian process with drift. This implies in particular that replacing the unknown bandwidth with a consistent estimate, leads to a survival function estimate that achieves the lower bound for the asymptotic MSE. We consider kernel estimation of the conditional survival function based on a sample of censored survival times and an explanatory variable. The approach requires only mild regularity conditions on the underlying model and can serve as an alternative to parametric or semiparametric regression models especially in exploratory data analysis or in the analysis of large data sets. Let X be a nonnegative random variable (r.v.) representing the survival time of an individual and let Z be a covariate such as age or blood pressure. The survival time X may be subject to right censoring in which case the observable r.v.'s are given by T = min (X, X?), & = I(T = X) and Z. Here XK is a nonnegative r.v. representing withdrawal time from the study. It is assumed that X and X? are independent conditionally on Z. Given an i.i.d. sample (Ti, Zi, bi) of such observations, we estimate the conditional survival function F(tjz) = P(X > tIZ = z) by the product integral estimate (Gill & Johansen, 1990)

Journal ArticleDOI
TL;DR: In this article, the central limit theorem for a class of U -statistics whose kernel depends on the sample size and for which the projection method may fail, since several terms in the Hoeffding decomposition contribute to the limiting variance.

Journal ArticleDOI
TL;DR: In this article, the central limit theorem and limit theorems for probabilities of large deviations for the solutions of such problems are proved for semilinear parabolic PDEs with fast oscillating boundary conditions.
Abstract: We consider semilinear parabolic PDE's with fast oscillating boundary conditions. The central limit theorem and limit theorems for probabilities of large deviations for the solutions of such problems are proved.


Journal ArticleDOI
TL;DR: In this paper, lower variance bounds for functions of a random vector X were derived, and the w -function associated with X was shown to characterize its distribution, and a special application showed the multivariate central limit theorem.

Journal ArticleDOI
TL;DR: In this paper, a description of the weak and strong limiting behaviour of weighted uniform tail empirical and tail quantile processes is given, and the results for the tail quantiles process are applied to obtain weak and strength functional limit theorems for a weighted non-uniform tail-quantile-type process based on a random sample from a distribution that satisfies the so called von Mises sufficient condition for being in the domain of max-attraction of a Frechet distribution.