Showing papers on "Central limit theorem published in 1982"

On the Central Limit Theorem for Stationary Mixing Random Fields

Abstract: A simple proof of a central limit theorem for stationary random fields under mixing conditions is given, generalizing some results obtained by more complicated methods, e.g. Bernstein's method.

Central Limit Theorems for Martingales with Discrete or Continuous Time

Abstract: This survey paper consists of two parts. In the first part (up to and including setion 3) we review the central limit theorems for discrete time martingales, and show that many different sets of conditions for convergence may be reduced to one basic set, where conditions are given in terms of conditional moments of truncated variables, given the past. In the second part (sections 4 and 5) we first recall some basic facts from the modern "French" theory of stochastic processes, then show that Rebolledo's recent functional limit theorems for martingales with continuous time can be deduced from the limit theorems for discrete time martingales. Again, several related sets of conditions for convergence are formulated.

A Central Limit Theorem for Stationary Processes and the Parameter Estimation of Linear Processes

Abstract: A central limit theorem is proved for the sample covariances of a linear process. The sufficient conditions for the theorem are described by more natural ones than usual. We apply this theorem to the parameter estimation of a fitted spectral model, which does not necessarily include the true spectral density of the linear process. We also deal with estimation problems for an autoregressive signal plus white noise. A general result is given for efficiency of Newton-Raphson iterations of the likelihood equation.

A central limit theorem for empirical processes

Abstract: The empirical measure Pn for independent sampling on a distribution P is formed by placing mass n−1 at each of the first n sample points. In this paper, n½(Pn − P) is regarded as a stochastic process indexed by a family of square integrable functions. A functional central limit theorem is proved for this process. The statement of this theorem involves a new form of combinatorial entropy, definable for classes of square integrable functions.

On Estimating the Endpoint of a Distribution

Abstract: We propose a method of estimating the endpoint, $\theta$, of a distribution when only limited information is available about the behaviour of the distribution in the neighbourhood of $\theta$. By using increasing numbers of extreme order statistics we obtain an estimator which improves on earlier estimators based on only a bounded number of extremes. In a certain particular model our estimator is equal to a maximum likelihood estimator, but it is robust against departures from this model.

Limit theorems for uniform distributions on spheres in high-dimensional euclidean spaces

Abstract: If X = (X 1, · ··, Xn ) has uniform distribution on the sphere or ball in ℝ with radius a, then the joint distribution of , ···, k, converges in total variation to the standard normal distribution on ℝ. Similar results hold for the inner products of independent n-vectors. Applications to geometric probability are given.

Small-sample asymptotic distributions of M-estimators of location

Abstract: SUMMARY Asymptotic formulae for the distribution of M-estimators, i.e. maximum likelihood type estimators, of location, including the arithmetic mean, are derived which numerical studies show to give relative errors for densities and tail areas of the order of magnitude of 1% down to sample sizes 3 and 4 even in the extreme tails. The paper is the continuation of earlier work by the second author and is also closely related to Daniels's work on the saddlepoint approximation. The method consists in expanding the derivative of the logarithm of the unstandardized density of the estimator in powers of 1/n at each point, using recentring by means of conjugate distributions. This method yields a unified point of view for the comparison of other asymptotic methods, namely saddlepoint method, Edgeworth expansion and large deviations approach, which are also compared numerically.

Nonlinear time series regression for a class of amplitude modulated consinusoids

Abstract: . A central limit theorem is proved for estimates of the unknown parameters in a time series which is a sum of amplitude modulated consinusoids observed subject to error. The amplitude function depends upon unknown parameters as well as the length of the series. The frequency for each cosinusoid is also assumed to be unknown. Estimates and standard errors are obtained through nonlinear least squares in the frequency domain.

A time series approach to the study of the simple subcritical galton-watson process

Abstract: The principal aim of this paper is to exhibit applications of techniques of time series analysis for establishing limit distribution theorems of statistical relevance on a subcritical Galton-Watson process X with immigration. In this approach the results obtained by Heyde and Seneta, Quine, and Klimko and Nelson are re-established in a more concise form on adopting new methods of proof, which seek to unify these results. In addition, Quenouille-type limit theorems on X are proved leading to the construction of Quenouille-type goodness-of-fit tests for X. It appears that Billingsley's central limit theorem for martingales is appropriate for proving the basic result, Theorem 1.1. This is done on converting the entire problem as a martingale problem through a use of Lemma 2 of Venkataraman (1968).

Limit theorems for sums of independent, compact, random subsets of euclidean space

Abstract: In the paper the asymptotic behavior of normalized, elementwise sums of independent, random, compact sets in Rm is studied. Criteria for the convergence of series of random sets are considered, and analogues of the central limit theorem and the strong law of large numbers are proved.

An almost sure invariance principle for Hilbert space valued martingales

Abstract: We obtain an almost sure approximation of a martingale with values in a real separable Hilbert space H by a suitable H-valued Brownian motion.

A time series approach to the study of the simple subcritical Galton–Watson process with immigration

Abstract: The principal aim of this paper is to exhibit applications of techniques of time series analysis for establishing limit distribution theorems of statistical relevance on a subcritical Galton–Watson process X with immigration. In this approach the results obtained by Heyde and Seneta, Quine, and Klimko and Nelson are re-established in a more concise form on adopting new methods of proof, which seek to unify these results. In addition, Quenouille-type limit theorems on X are proved leading to the construction of Quenouille-type goodness-of-fit tests for X. It appears that Billingsley's central limit theorem for martingales is appropriate for proving the basic result, Theorem 1.1. This is done on converting the entire problem as a martingale problem through a use of Lemma 2 of Venkataraman (1968).

A chaos hypothesis for some large systems of random equations

Abstract: This paper is about the behavior of solutions to large systems of linear algebraic and differential equations when the coefficients are random variables. We will prove a law of large numbers and a central limit theorem for the solutions of certain algebraic systems, and the weak convergence to a Gaussian process for the solution of a system of differential equations. Some of the results were surprisingly difficult to prove, but they are all easily anticipated from a “chaos hypothesis”: i.e. an assumption of near independence for the components of the solutions of large systems of weakly coupled equations.

On the Asymptotic Behavior of Random Walks on an Anisotropic Lattice

Abstract: A random walk on a two-dimensional lattice with homogeneous rows and inhomogeneous columns, which could serve as a model for the study of some transport phemonema, is discussed. Subject to an asymptotic density condition on the columns it is shown that the horizontal motion of the walk is asymptotically like that of rescaled Brownian motion. Various consequences of this are derived including central limit, iterated logarithm, and mean square displacement results for the horizontal component of the walk.

Branching random walk in varying environments

Abstract: The branching random walk model is generalized towards generationdependent displacement and reproduction distributions. Asymptotic theory of branching random walk in varying environments from the L2 point of view is given. If Z,(x) is the number of nth-generation particles to the left of x, then under appropriate conditions for suitably chosen x,, Z, (x,)/Z,(+oo) converges in L2 completely to a limiting distribution. Sufficient conditions for almost sure convergence are given. As a corollary an analogue of the central limit theorem for the proportion of particles of the nth generation in time interval I, in the age-dependent Crump-Mode-Jagers process is obtained.

Probability distributions of the Kolmogorov and omega-square statistics for continuous distributions with shift and scale parameters

Abstract: The paper is devoted to the application of the statistical Kolmogorov and omega-square criteria to verification of a complex hypothesis H0 according to which the independent, identically and continuously distributed random variables X1,...,Xn have the law G[(x−θ 1)/θ 2].

Bounds on the Rate of Convergence of Moments in the Central Limit Theorem

Abstract: We derive two-sided bounds on the rate of convergence of moments in the central limit theorem. A variety of norming constants is considered, and it is shown that a very delicate alteration to these constants can have a significant effect on the rate. Indeed, the influence of norming constants on rates of convergence of moments is of a more subtle nature than in the central limit theorem itself. We present several examples to illustrate some extreme cases.

A Berry-Esseen Bound for an Occupancy Problem

Abstract: A Berry-Esseen bound is given for the rate of convergence to normality of the number of empty boxes when balls are distributed independently and at random to boxes with possibly unequal probabilities The method of proof uses the equivalence of this distribution to a certain conditional distribution based on independent Poisson random variables Then methods based on the characteristic function of this conditional distribution are used to obtain the result

Chromatographic Kinetics and the Phenomenon of Tailing

Abstract: This paper contains a reformulation of the kinetic equations that describe transport in a chromatographic column, to allow the possibility of describing adsorption at different types of sites. It is shown that in place of the partial differential equation that is the usual starting point for any analysis, one obtains a partial integro-differential equation. While only a formal solution to this equation is possible, the central limit theorem of probability guarantees that in a wide variety of cases the solution for an isolated peak will approach the Gaussian form, so that (∗) is applicable. Next the conditions necessary to insure the appearance of tailing or permanent asymmetry in an isolated peak are considered. Two models that lead to tailing are considered. In the first there is a single adsorption phase, but the residence time distribution has an infinite variance. The second consists of a random distribution of first-order rate constants.