Showing papers on "Central limit theorem published in 1984"

Convergence of stochastic processes

Abstract: I Functional on Stochastic Processes.- 1. Stochastic Processes as Random Functions.- Notes.- Problems.- II Uniform Convergence of Empirical Measures.- 1. Uniformity and Consistency.- 2. Direct Approximation.- 3. The Combinatorial Method.- 4. Classes of Sets with Polynomial Discrimination.- 5. Classes of Functions.- 6. Rates of Convergence.- Notes.- Problems.- III Convergence in Distribution in Euclidean Spaces.- 1. The Definition.- 2. The Continuous Mapping Theorem.- 3. Expectations of Smooth Functions.- 4. The Central Limit Theorem.- 5. Characteristic Functions.- 6. Quantile Transformations and Almost Sure Representations.- Notes.- Problems.- IV Convergence in Distribution in Metric Spaces.- 1. Measurability.- 2. The Continuous Mapping Theorem.- 3. Representation by Almost Surely Convergent Sequences.- 4. Coupling.- 5. Weakly Convergent Subsequences.- Notes.- Problems.- V The Uniform Metric on Spaces of Cadlag Functions.- 1. Approximation of Stochastic Processes.- 2. Empirical Processes.- 3. Existence of Brownian Bridge and Brownian Motion.- 4. Processes with Independent Increments.- 5. Infinite Time Scales.- 6. Functional of Brownian Motion and Brownian Bridge.- Notes.- Problems.- VI The Skorohod Metric on D(0, ?).- 1. Properties of the Metric.- 2. Convergence in Distribution.- Notes.- Problems.- VII Central Limit Theorems.- 1. Stochastic Equicontinuity.- 2. Chaining.- 3. Gaussian Processes.- 4. Random Covering Numbers.- 5. Empirical Central Limit Theorems.- 6. Restricted Chaining.- Notes.- Problems.- VIII Martingales.- 1. A Central Limit Theorem for Martingale-Difference Arrays.- 2. Continuous Time Martingales.- 3. Estimation from Censored Data.- Notes.- Problems.- Appendix A Stochastic-Order Symbols.- Appendix B Exponential Inequalities.- Notes.- Problems.- Appendix C Measurability.- Notes.- Problems.- References.- Author Index.

Asymptotic theory for econometricians

Abstract: The Linear Model and Instrumental Variables Estimators. Consistency. Laws of Large Numbers. Asymptotic Normality. Central Limit Theory. Estimating Asymptotic Covariance Matrices. Functional Central Limit Theory and Applications. Directions for Further Study. Solution Set. References. Index.

Some Limit Theorems for Empirical Processes

Abstract: In this paper we provide a general framework for the study of the central limit theorem (CLT) for empirical processes indexed by uniformly bounded families of functions $\mathscr{F}$ From this we obtain essentially all known results for the CLT in this case; we improve Dudley's (1982) theorem on entropy with bracketing and Kolcinskii's (1981) CLT under random entropy conditions One of our main results is that a combinatorial condition together with the existence of the limiting Gaussian process are necessary and sufficient for the CLT for a class of sets (modulo a measurability condition) The case of unbounded $\mathscr{F}$ is also considered; a general CLT as well as necessary and sufficient conditions for the law of large numbers are obtained in this case The results for empiricals also yield some new CLT's in $C\lbrack 0, 1\rbrack$ and $D\lbrack 0, 1\rbrack$

Multivariate linear time series models

Abstract: This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree, n, because of the fact that this space, M(n), can be realised as a kind of high-dimensional algebraic surface of dimension n(2s + m) where s and m are the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determining n and then the appropriate element of M(n). However, the fact that M(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2. The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data. The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record, T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.

Limit Theorems for Certain Diffusion Processes with Interaction

Abstract: Publisher Summary This chapter discusses the limit theorems for certain diffusion processes with interaction. The chapter studies the central limit theorem and the large deviation problem for U n in the case when b(x, y) is general but σ = 1, by amplifying the method of Braun and Hepp. The advantage of the present method is that the result in the path space can be easily obtained and also the I -functional governing the large deviation for Un can find. The chapter discusses a method of Bran and Hepp. A general theorem, which is an abstraction of the method used by Braun and Hepp is stated in the proof. The chapter discusses the central limit theorem. The solution ξ(t) of (σ = 1) is an inhomogeneous Markov process. This Markov process is called the diffusion process associated with the nonlinear parabolic equation(σ = 1).

A Berry-Esseen bound for symmetric statistics

Abstract: The rate of convergence of the distribution function of a symmetric function of N independent and identically distributed random variables to its normal limit is investigated. Under appropriate moment conditions the rate is shown to be (\(O\left( {{N^{ - \frac{1}{2}}}} \right)\)). This theorem generalizes many known results for special cases and two examples are given. Possible further extensions are indicated.

Limit theorems for sums of general functions of m -spacings

Abstract: Laws of large numbers and central limit theorems are proved for sums of general functions of m-spacings from general distributions. Explicit formulae are given for the norming constants. The results enable us to describe asymptotic properties of distributional tests under fixed alternatives. A generalization of Kimball's spacings test is considered in detail.

Poisson Convergence for Dissociated Statistics

Abstract: SUMMARY A Poisson limit theorem is derived for the number of "large" values observed among comparisons of independent, but not necessarily identically distributed random variables. The comparisons made need not be the same and may depend on the two variables being compared. An application to the assessment of large numbers of correlation coefficients is given.

Limit theorems in probability and statistics

Abstract: Some Hitting Probabilities of Random Walks on Z 2 (P. Auer). On the Relative Frequency of Points Visited by Random Walk on Z 2 (P. Auer, P. Revesz). Probability Theory of the Trigonometric System (I. Berkes). Counterexamples to the Central Limit Theorem under Strong Mixing Conditions, II (R.C. Bradley). Bootstrapped Parameter Estimated Quantile Processes L-Estimation Case (M.D. Burke, E. Gombay). A Liminf Result in Strassen's Law of the Iterated Logarithm (E. Csaki). On Confidence Bands for the Quantile Function of a Continuous Distribution Function (M. Csorgo, L. Horvath). Path Properties of Infinite Dimensional Ornstein-Uhlenbeck Processes (M. Csorgo, Z.Y. Lin). Asymptotic Distributions for Vectors of Power Sums (S. Csorgo, L. Viharos). A Bahadur-Kiefer-Type Two Sample Statistic with Applications to Tests of Goodness of Fit (P. Deheuvels, D.M. Mason). Asysmptotic Normality of Semi-Nonparametric Estimators Using Minimization of Cp or Adaptive Polynomial Truncation Selection Procedures (B.J. Eastwood). Some Results on the Almost Sure Behavior of Martingales (U. Einmahl, D.M. Mason). On Some Limit Theorems for Weakly Multiplicative Systems (K. Fukuyama). Strong Approximation Theorems for Estimator Processes in Continuous Time (L. Gerencser). An Entropy Estimate Based on a Kernel Density Estimation (L. Gyorfi, E.C. van der Meulen). A Law of the Iterated Logarithm for Modulus Trimming (E. Haeusler, D.M. Mason). On Whittle's Approximation in Spectral Analysis of Stationary Gaussian Time Series (K. Hornik). Limiting Distributions of Poisson Bridges, Kendall-Kendall Pontograms and Related Processes (V.R. Huse-Eastwood). A Normal Convergence Criterion for Strongly Mixing Stationary Sequences (A. Jakubowski, Z. Szewczak). Asymptotic Normality of Linear-Quadratic Statistics for Gaussian Random Fields (M. Janzura). A Diffusion Approximation for the Quantization Error in Delta Modulation with a Gauss-Markov Signal (T. Koski). The Convergence of Moments in the Martingale Central Limit Theorem (K.S. Kubacki). Weak Convergence of Different Types of Variation for Biparametric Gaussian Processes (J.R. Leon, J. Ortega). Cumulant Spectral Estimates: Bias and Covariance (K.S. Lii, M. Rosenblatt). On the Local Limit Problem for Densities (W. Macht, W. Wolf). A Discrete-Time Analogue of the M/M/1 Queue and the Transient Solution: An Analytic Approach (S.G. Mohanty, W. Panny). A.S. Behavior of the Cesaro Means for Quasi-Orthogonal Random Fields (F. Moricz). Multidimensional Domains of Large Deviations (W.D. Richter). Optimality of BLUE and ABLUE in the Light of the Pitman Closeness of Statistical Estimators (P.K. Sen). The Gamma Process and the Poisson Distribution (F.W. Steutel, J.G.F. Thiemann). A Discrete Ito's Formula (T. Szabados). On the Asymptotic Efficiency of Estimates of Parameters of Regular Densities (D. Vorlickova). Some Limit Results of the Local Time of Wiener Process and Random Walk (Z. Yong).

Large deviations for Poisson systems of independent random walks

Abstract: We prove large deviation theorems for occupation time functionals of independent random walks started from a Poisson field on Z d. In dimensions 1 and 2 the large deviation tails are larger than exponential. Exact asymptotics are derived.

The central limit theorem for the Poisson shot-noise process

Abstract: The Poisson shot-noise process discussed here takes the form f:oo H(t, s)N(ds), where N(· is the counting measure of a Poisson process and the H(·, s) are independent stochastic processes. Necessary and sufficient conditions are obtained for convergence in distribution, as t ∼ OC, to any infinitely divisible distribution. The main interest is in the explosive transient one-sided shot-noise, Y(t) = f:1 H(t, s)N(ds) where Var Y(t)∼ oc, Here conditions for asymptotic normality are discussed in detail. Important examples include the Poisson cluster point process and the integrated stationary shotnoise.

Integrated Square Error Properties of Kernel Estimators of Regression Functions

Abstract: Weak laws of large numbers and central limit theorems are proved for integrated square error of kernel estimators of regression functions. The regressor is assumed to take values in $\mathbb{R}^p$, and the regressand, $X$, to be real valued. It is shown that in many cases, integrated square error is asymptotically normally distributed and independent of the $X$-sample. As an application, a test for the regression function (such as that proposed by Konakov) is seen to be asymptotically independent of an arbitrary test based on the $X$-sample. The proofs involve martingale methods.

Limit Theorems for Sums of Independent Random Variables Defined on a Recurrent Random Walk

Abstract: Let {Xi}i=∞ -∞, {ξi}i=1∞ be two independent sequences of randoms variables, where the ξi are identically distributed and assume integer values. Let In the paper the question of the asymptotic behavior as n → ∞ of the quantity is considered. It is shown that the distribution of Wn converges to the distribution of the normal law and that the estimate of the rate of convergence has the same order as the classical estimate of Berry-Esseen.

Parametric representation of mean values for stationary random mosaies

Abstract: Mean values referring to stationary random 2- or 3-dimensional, mosaics are considered, e.g. the mean number of vertices per unit volume, the mean content of the typical cell, the mean perimeter of the typical face, the mean number of emanating edges of the typical vertex etc, A parametric representation of some mean values is given in order to explain the relations between them, The parameters of the superposition of two independent isotropic random mosaics are evaluated

Central limit theorem for products of random matrices

Abstract: Using the semigroup product formula of P Chernoff, a central limit theorem is derived for products of random matrices Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter product formula

Persistent random walks in a one-dimensional random environment

Abstract: Central limit theorems are obtained for persistent random walks in a onedimensional random environment. They also imply the central limit theorem for the motion of a test particle in an infinite equilibrium system of point particles where the free motion of particles is combined with a random collision mechanism and the velocities can take on three possible values.

Variance reduction in queueing simulation using generalized concomitant variables

Abstract: To improve the efficiency of system performance estimators generated by a queueing simulation, procedures are developed for exploiting standardized concomitant variables that are associated with each input process sampled during the simulation. A multivariate central limit theorem is established for such variables in a broad class of regenerative queueing systems. This result is the basis for robust, asymptotically stable variance reduction techniques that incorporate these concomitant variables into poststratified sampling schemes as well as control-variate schemes. Each procedure is adapted to estimation methods based on replication analysis and regenerative analysis. A summary of the results of an experimental performance evaluation indicates the potential efficiency gains that can be achieved with these procedures

Functional law of the iterated logarithm and uniform central limit theorem for partial-sum processes indexed by sets

Abstract: Let $\{X_j: \mathbf{j} \in J^d\}$ be an array of independent random variables, where $J^d$ denotes the $d$-dimensional positive integer lattice. The main purpose of this paper is to obtain a functional law of the iterated logarithm (LIL) for suitably normalized and smoothed versions of the partial-sum process $S(B) = \sum_{j \in B}X_j$. The method of proof involves the definition of a set-indexed Brownian process, and the embedding of the partial-sum process in this Brownian process. In addition, the LIL is derived for this Brownian process. The method is extended to yield a uniform central limit theorem for the partial-sum process.

Non-Uniform Estimates and Asymptotic Expansions of the Remainder in the Central Limit Theorem for m-Dependent Random Variables

Abstract: An intracorporal drive, and method of operating the same, especially for an extension unit for extension osteotomy and for a compression unit for pressure osteosynthesis, wherein the driving power is generated by the osmotic pressure between two differently concentrated solutions separated from each other by a semipermeable diaphragm or membrane. The solution of low concentration also can be substituted by pure solvent.

Parameter Estimation of Stationary Processes with Spectra Containing Strong Peaks

Abstract: The advantage of using data tapers in parameter estimation of stationary processes are investigated. Consistency and a central limit theorem for quasi maximum likelihood parameter estimates with tapered data are proved, and data tapers leading to asymptotically efficient parameter estimates are determined. Finally the results of a Monte Carlo study on Yule-Walker estimates with tapered data are presented.

Expansions for von Mises functionals

Abstract: Berry-Esseen results and expansions are derived for the distribution function of von Mises functionals of order r under moment conditions and conditions on the smoothness of the limit distribution. The results apply to goodness-of-fit statistics — as well as to the central limit theorem in L 2p,p≧2, the rate of convergence being O(n −1) for centered balls, provided a fourth moment exists.