Showing papers on "Central limit theorem published in 1986"

Asymptotic methods in statistical decision theory

Abstract: 1 Experiments-Decision Spaces.- 1 Introduction.- 2 Vector Lattices-L-Spaces-Transitions.- 3 Experiments-Decision Procedures.- 4 A Basic Density Theorem.- 5 Building Experiments from Other Ones.- 6 Representations-Markov Kernels.- 2 Some Results from Decision Theory: Deficiencies.- 1 Introduction.- 2 Characterization of the Spaces of Risk Functions: Minimax Theorem.- 3 Deficiencies Distances.- 4 The Form of Bayes Risks-Choquet Lattices.- 3 Likelihood Ratios and Conical Measures.- 1 Introduction.- 2 Homogeneous Functions of Measures.- 3 Deficiencies for Binary Experiments: Isometries.- 4 Weak Convergence of Experiments.- 5 Boundedly Complete Experiments.- 6 Convolutions: Hellinger Transforms.- 7 The Blackwell-Sherman-Stein Theorem.- 4 Some Basic Inequalities.- 1 Introduction.- 2 Hellinger Distances: L1-Norm.- 3 Approximation Properties for Likelihood Ratios.- 4 Inequalities for Conditional Distributions.- 5 Sufficiency and Insufficiency.- 1 Introduction.- 2 Projections and Conditional Expectations.- 3 Equivalent Definitions for Sufficiency.- 4 Insufficiency.- 5 Estimating Conditional Distributions.- 6 Domination, Compactness, Contiguity.- 1 Introduction.- 2 Definitions and Elementary Relations.- 3 Contiguity.- 4 Strong Compactness and a Result of D. Lindae.- 7 Some Limit Theorems.- 1 Introduction.- 2 Convergence in Distribution or in Probability.- 3 Distinguished Sequences of Statistics.- 4 Lower-Semicontinuity for Spaces of Risk Functions.- 5 A Result on Asymptotic Admissibility.- 8 Invariance Properties.- 1 Introduction.- 2 The Markov-Kakutani Fixed Point Theorem.- 3 A Lifting Theorem and Some Applications.- 4 Automatic Invariance of Limits.- 5 Invariant Exponential Families.- 6 The Hunt-Stein Theorem and Related Results.- 9 Infinitely Divisible, Gaussian, and Poisson Experiments.- 1 Introduction.- 2 Infinite Divisibility.- 3 Gaussian Experiments.- 4 Poisson Experiments.- 5 A Central Limit Theorem.- 10 Asymptotically Gaussian Experiments: Local Theory.- 1 Introduction.- 2 Convergence to a Gaussian Shift Experiment.- 3 A Framework which Arises in Many Applications.- 4 Weak Convergence of Distributions.- 5 An Application of a Martingale Limit Theorem.- 6 Asymptotic Admissibility and Minimaxity.- 11 Asymptotic Normality-Global.- 1 Introduction.- 2 Preliminary Explanations.- 3 Construction of Centering Variables.- 4 Definitions Relative to Quadratic Approximations.- 5 Asymptotic Properties of the Centerings $$\hat{Z}$$.- 6 The Asymptotically Gaussian Case.- 7 Some Particular Cases.- 8 Reduction to the Gaussian Case by Small Distortions.- 9 The Standard Tests and Confidence Sets.- 10 Minimum ?2 and Relatives.- 12 Posterior Distributions and Bayes Solutions.- 1 Introduction.- 2 Inequalities on Conditional Distributions.- 3 Asymptotic behavior of Bayes Procedures.- 4 Approximately Gaussian Posterior Distributions.- 13 An Approximation Theorem for Certain Sequential Experiments.- 1 Introduction.- 2 Notations and Assumptions.- 3 Basic Auxiliary Lemmas.- 4 Reduction Theorems.- 5 Remarks on Possible Applications.- 14 Approximation by Exponential Families.- 1 Introduction.- 2 A Lemma on Approximate Sufficiency.- 3 Homogeneous Experiments of Finite Rank.- 4 Approximation by Experiments of Finite Rank.- 5 Construction of Distinguished Sequences of Estimates.- 15 Sums of Independent Random Variables.- 1 Introduction.- 2 Concentration Inequalities.- 3 Compactness and Shift-Compactness.- 4 Poisson Exponentials and Approximation Theorems.- 5 Limit Theorems and Related Results.- 6 Sums of Independent Stochastic Processes.- 16 Independent Observations.- 1 Introduction.- 2 Limiting Distributions for Likelihood Ratios.- 3 Conditions for Asymptotic Normality.- 4 Tests and Distances.- 5 Estimates for Finite Dimensional Parameter Spaces.- 6 The Risk of Formal Bayes Procedures.- 7 Empirical Measures and Cumulatives.- 8 Empirical Measures on Vapnik-?ervonenkis Classes.- 17 Independent Identically Distributed Observations.- 1 Introduction.- 2 Hilbert Spaces Around a Point.- 3 A Special Role for $$\sqrt{n}$$: Differentiability in Quadratic Mean.- 4 Asymptotic Normality for Rates Other than $$\sqrt{n}$$.- 5 Existence of Consistent Estimates.- 6 Estimates Converging at the $$\sqrt{n}$$-Rate.- 7 The Behavior of Posterior Distributions.- 8 Maximum Likelihood.- 9 Some Cases where the Number of Observations Is Random.- Appendix: Results from Classical Analysis.- 1 The Language of Set Theory.- 2 Topological Spaces.- 3 Uniform Spaces.- 4 Metric Spaces.- 5 Spaces of Functions.- 6 Vector Spaces.- 7 Vector Lattices.- 8 Vector Lattices Arising from Experiments.- 9 Lattices of Numerical Functions.- 10 Extensions of Positive Linear Functions.- 11 Smooth Linear Functionals.- 12 Derivatives and Tangents.

Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions

Abstract: We prove a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains under virtually no assumptions other than the necessary ones. We use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk.

Entropy and the Central Limit Theorem

Abstract: A strengthened central limit theorem for densities is established showing monotone convergence in the sense of relative entropy

Limit Theorems for Fuzzy Random Variables

Abstract: A strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables, whose values are fuzzy sets with compact levels. The proofs are based on embedding theorems as well as on probability techniques in Banach space.

A weighted occupation time for a class of measured-valued branching processes

Abstract: A weighted occupation time is defined for measure-valued processes and a representation for it is obtained for a class of measure-valued branching random motions on Rd. Considered as a process in its own right, the first and second order asymptotics are found as time t→∞. Specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d, and when infinite, a central limit type renormalization is considered, yielding Gaussian or asymmetric stable generalized random fields in the limit. In one Gaussian case the results are contrasted in high versus low dimensions.

Probability: An Introduction

Abstract: This new undergraduate text offers a concise introduction to probability and random processes. Exercises and problems range from simple to difficult, and the overall treatment, though elementary, includes rigorous mathematical arguments. Chapters contain core material for a beginning course in probability, a treatment of joint distributions leading to accounts of moment-generating functions, the law of large numbers and the central limit theorem, and basic random processes.

The Central Limit Theorem Around 1935

Abstract: A long standing problem of probability theory has been to find necessary and sufficient conditions for the approximation of laws of sums of random variables by Gaussian distributions. A chapter in that search was closed by the 1935 work of Feller and Levy and by a beautiful result of Cramer published in early 1936. We review the respective contributions of Feller and Levy mentioning as necessary contributions of Laplace, Poisson, Lindeberg, Bernstein, Kolmogorov, and others, with an effort to place them in the context of the authors' times and in a modern content.

Propriétés d'intersection des marches aléatoires. I. Convergence vers le temps local d'intersection

Abstract: We study intersection properties of multi-dimensional random walks. LetX andY be two independent random walks with values in ℤd (d≦3), satisfying suitable moment assumptions, and letIn denote the number of common points to the paths ofX andY up to timen. The sequence (In), suitably normalized, is shown to converge in distribution towards the “intersection local time” of two independent Brownian motions. Results are applied to the proof of a central limit theorem for the range of a two-dimensional recurrent random walk, thus answering a question raised by N. C. Jain and W. E. Pruitt.

Recent Advances in the Central Limit Theorem and Its Weak Invariance Principle for Mixing Sequences of Random Variables (A Survey)

Abstract: The purpose of this paper is to describe the progress that has recently been made in the study of the central limit theorem and its weak invariance principle for mixing sequences of random variables and to point out some open problems in this subject.

Central Limit Theorems for Infinite Series of Queues and Applications to Simple Exclusion

Abstract: We prove that a tagged particle in asymmetric simple exclusion satisfies a central limit theorem when properly rescaled. To obtain this result we derive several results of positive (or negative) correlation for occupation times of a server in a series of queues which imply various central limit theorems.

What portion of the sample makes a partial sum asymptotically stable or normal

Abstract: Let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0<α≦2 be given. We show that if at each stage n a number k n depending on n of the lower and upper order statistics are removed from the n-th partial sum of the given random variables then under appropriate conditions on k n the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.

Law of Large Numbers and Central Limit Theorem for Linear Chemical Reactions with Diffusion

Abstract: Two mathematical models of chemical reactions with diffusion for a single reactant in a one-dimensional volume are compared, namely, the deterministic and the stochastic models. The deterministic model is given by a partial differential equation, the stochastic one by a space-time jump Markov process. By the law of large numbers the consistency of the two models is proved. The deviation of the stochastic model from the deterministic model is estimated by a central limit theorem. This limit is a distribution-valued Gauss-Markov process and can be represented as the mild solution of a certain stochastic partial differential equation.

Invariance Principles for Independent and Weakly Dependent Random Variables

Abstract: During the last three decades the concept “invariance principle” has undergone several subtle changes. In the early 1950’s an invariance principle was a result that nowadays often would be called a functional central limit theorem (FCLT). At present the term “invariance principle” generally stands as a synonym for an approximation theorem: A given process, such as a partial sum process, an empirical process, an extremal process, a U-statistic, etc. is approximated in distribution, in probability, in LP or almost surely by a canonical process, such as a Brownian motion, a Kiefer process, a special extremal process or in case of a U-statistic by a multiple stochastic integral.

Asymptotic expansions based on smooth functions in the central limit theorem

Abstract: Stein's method is used to derive asymptotic expansions for expectations of smooth functions of sums of independent random variables, together with Lyapounov estimates of the error in the approximation.

An asymptotically efficient difference formula for solving stochastic differential equations

Abstract: Time discretisations of teh vector stochastic differential equation are considered, where (y t) is a continuous scalar process whose distribution is absolutely continuous with respect to Wiener measure. Among approximations to x T that depend on (y t) only at the discretisation points, the conditional mean is asymptotically optimal in the sense that it minimises all symmetrical conditional moments of the error. A conditional central limit theorem is derived for this minimal error, and a finite difference formula is developed which yields appromimations with the same asymptotically optimal properties. This formula is necessarily more complex than the familiar Milshtein scheme (Milshtein [7]). The latter has the maximum order of convergence but its error, considered as a power serioes in the discretisation parameter h, does not have the minimal leading coefficient. The results generalise for a special class of equations with multi-dimensional forcing terms

A central-limit-theorem version of L =λ W

Abstract: Underlying the fundamental queueing formulaL=λW is a relation between cumulative processes in continuous time (the integral of the queue length process) and in discrete time (the sum of the waiting times of successive customers). Except for remainder terms which usually are asymptotically negligible, each cumulative process is a random time-transformation of the other. As a consequence, in addition to the familiar relation between the with-prob ability-one limits of the averages, roughly speaking, the customer-average wait obeys a central limit theorem if and only if the time-average queue length obeys a central limit theorem, in which case both averages, properly normalized, converge in distribution jointly, and the individual limiting distributions are simply related. This relation between the central limit theorems is conveniently expressed in terms of functional central limit theorems, using the continuous mapping theorem and related arguments. The central limit theorems can be applied to compare the asymptotic efficiency of different estimators of queueing parameters. For example, when the arrival rateλ is known and the interarrivai times and waiting times are negatively correlated, it is more asymptotically efficient to estimate the long-run time-average queue lengthL indirectly by the sample-average of the waiting times, invokingL=λW, than it is to estimate it by the sample-average of the queue length. This variance-reduction principle extends a corresponding result for the standard GI/G/s model established by Carson and Law [2].

Rates of convergence in the central limit theorem for empirical processes

Abstract: In this paper we study we uniform behavior of the empirical brownian bridge over families of functions F bounded by a function F (the observations are independent with common distribution P). Under some suitable entropy conditions which were already used by Kolcinskii and Pollard, we prove exponential inequalities in the uniformly bounded case where F is a constant (the classical Kiefer's inequality (1961) is improved), as well as weak and strong invariance principles with rates of convergence in the case where F belongs to L2+δ(P) with δe]0,1] (our results improve on Dudley, Philipp's results (1983) whenever F is a Vapnik-Cervonenkis class in the uniformly bounded case and are new in the unbounded case).

Uniform Integrability and the Central Limit Theorem for Strongly Mixing Processes

Abstract: It is shown that for a strongly mixing sequence the central limit theorem holds if and only if the squares of the normalised partial sums are uniformly integrable.

Periodic behavior of the stochastic Brusselator in the mean-field limit

Abstract: We prove a “propagation of chaos” result for the mean-field limit of a model for a trimolecular chemical reaction called “Brusselator”. Then we show that the pair of nonlinear (i.e. law-dependent) stochastic differential equations describing the evolution of the concentration of the molecules at a given site in the mean field limit has a solution with a periodic law (in t). Finally we identify the limit and establish a central limit theorem for the periodic law in the case where the noise tends to zero.

On the central limit theorem in Rp when p

Abstract: Let X 1 , X 2 , ..., X n be i.i.d. random vectors in R p where p tends to infinity. A theorem is presented showing that the Central Limit Theorem should hold if p 2/n tends to zero. Furthermore, an example is presented with X i having a mixed multivariate normal distribution (with finite moment generating function) for which a uniform normal approximation to the distribution of the sample mean $$(\sqrt {\text{n}} \overline {\text{X}} )$$ can not hold if p 2/n does not tend to zero.

The asymptotic number of acyclic digraphs

Abstract: We obtain an asymptotic formula forA n,q , the number of digraphs withn labeled vertices,q edges and no cycles. The derivation consists of two separate parts. In the first we analyze the generating function forA n,q so as to obtain a central limit theorem for an associated probability distribution. In the second part we show combinatorially thatA n,q is a smooth function ofq. By combining these results, we obtain the desired asymptotic formula.

Central‐limit theorems on groups

Abstract: The probability density pN of the product of N statistically independent (and identically distributed, each with probability density p1) elements of a group is studied in the limit N→∞. It is shown, for the compact groups R(2) and R(3), that pN→1 as N→∞, independently of p1. It is made plausible that a similar behavior is to be expected for other compact groups. For noncompact groups, the case of SU(1,1)which is of interest to the physics of disordered conductors, is studied. The case in which p1 is isotropic, i.e., independent of the phases, is analyzed in detail. When p1 is fixed and N≫1, a Gaussian distribution in the appropriate variable is found. When the original variables are rescaled by 1/N and the limit N→∞ is taken, keeping the ratio of the length of the conductor to the localization length fixed, an explicit integral representation for the resulting probability density is found. It is also exhibited that the latter satisfies a ‘‘diffusion’’ equation on the group manifold.

Computation of the chi square and Poisson distribution

Abstract: The paper deals with the computation of upper and lower tail probabilities of the chi-square and Poisson distribution with a specified relative accuracy on both tails for virtually all possible parameter values. With some supplement the proposed algorithms will also work for the general gamma distribution. If the parameters are small, open forward and backward recursion is used for the summation with an adaptive number of steps depending on the specified accuracy. For large parameters asymptotic expansions related to the central limit theorem are applied for the approximation. The basic ideas of the proposed methods will also be applicable to other elementary statistical distributions such as the binomial, beta, and F-distribution as well as the hypergeometric distribution.

Bahadur Representations for Robust Scale Estimators Based on Regression Residuals

Abstract: We investigate the asymptotic behaviour of the median deviation and the semi-interquartile range based on the residuals from a linear regression model by deriving weak asymptotic representations for the estimators. These representations may be used to obtain a variety of central limit theorems and yield conditions under which the median deviation and the semi-interquartile range are asymptotically equivalent. The results justify the use of the estimators as concommitant scale estimators in the general scale equivariant M-estimation of a regression parameter problem. Finally, the results contain as a special case those obtained by Hall and Welsh (1985) for independent and identically distributed random variables.

On the Rate of Convergence in the Central Limit Theorem in Banach Spaces

Abstract: Let $E$ denote a separable Banach space and let $X_i, i \in \mathbb{N}$, be a sequence of i.i.d. $E$-valued random vectors having finite third moment such that the central limit theorem holds. We prove that the convergence rate in the central limit theorem is $O(n^{-1/2})$ for regions $\{x \in E: F(x) < r\}$ which are defined by means of a smooth real valued function $F$ on $E$, provided that the limiting distribution of the gradient of $F$ fulfills a variance condition. Using this result we prove that the rate of convergence in the functional limit theorem for empirical processes is of order $O(n^{-1/2})$.

Remarks on the central limit theorem for weakly dependent random variables

Abstract: to a normal law. A sequence {mi]i~ ~ of random variables adapted to some increasing family of ~-algebras (~}ia~ are called martingale differences if E(mi+11~) = O for all i. The first one to observe that the standard results for CLT's for weakly dependent random variables may be obtained from the CLT for martingale differences was Gordin [3]. This approach was then taken up by others (see the monograph by Hall and Heyde and references therein