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Showing papers on "Central limit theorem published in 1977"


Journal ArticleDOI
TL;DR: The ratio-of-uniforms method for generating random variables having continuous nonuniform distributions is presented and can be used for generating short and often as fast algorithms as well as longer algorithms.
Abstract: The ratio-of-uniforms method for generating random variables having continuous nonuniform distributions is presented. In thin method a point is generated uniformly over a particular region of the plane. The ratio of the coordinate values of thin point yields a deviate with the desired distribution. Algorithms which utilize this techmque are generally short and often as fast as longer algorithms.

207 citations


Journal ArticleDOI
TL;DR: In this article, the limiting behavior of the critical measure diffusion process is investigated and conditions are found under which a non-trivial steady state random measure exists and in this case a spatial central limit theorem is established.
Abstract: A multiplicative stochastic measure diffusion process is the continuous analogue of an infinite particle branching diffusion process. In this paper the limiting behavior of the critical measure diffusion process is investigated. Conditions are found under which a non-trivial steady state random measure exists and in this case a spatial central limit theorem is established.

145 citations


Journal ArticleDOI
TL;DR: In this paper, the authors used Gastwirth's definition of the Lorenz curve which applies to a general probability distribution on (0, oo) having finite mean; thus it applies both to a 'population' distribution Y and to empirical distributions obtained on sampling.
Abstract: The Lorenz curve of the distribution of 'wealth' is a graph of cumulative proportion of total 'wealth' owned, against cumulative proportion of the population owning it. This paper uses Gastwirth's definition of the Lorenz curve which applies to a general probability distribution on (0, oo) having finite mean; thus it applies both to a 'population' distribution Y and to empirical distributions obtained on sampling. The Lorenz curves of the latter are proved to converge, with probability 1, uniformly to the former, and similarly for their inverses. Modified Lorenz curves are also defined, which treat atoms of 9Y differently, and these and their inverses are proved strongly consistent. Functional central limit theorems are then proved for empirical Lorenz curves and their inverses, under condition that Y be continuous and have finite variance. A mild variation condition is also needed in some circumstances. If the support of P is connected, the weak convergence is relative to C[0, 1] with uniform topology, otherwise to D [0, 1] with M1 topology. Selected applications are discussed, one being to the Gini coefficient.

123 citations


Journal ArticleDOI
TL;DR: In this paper, the authors developed in a more general setting the methods used' by Paulson, Holcomb & Leitch (1975) to estimate the parameters of a stable law, and established consistency under the condition of differentiability of the characteristio function and the existence of bounded second derivatives.
Abstract: SUMMARY The paper develops in a more general setting the methods used' by Paulson, Holcomb & Leitch (1975) to estimate the parameters of a stable law The statistic considered minimizes a distance function determined, by the empirical characteristic function Consistency is established under the condition of differentiability of the characteristio function and the existence of bounded second derivatives is required to obtain a central limit theorem for the estimators of one or more parameters Questions concerning efficiency and robustness are discussed

113 citations


Journal ArticleDOI
TL;DR: In this article, a simple expression for the characteristic functional of generalized shot noise is developed and a central limit theorem for moment functions of all orders is derived, and several examples are discussed.
Abstract: A simple expression for the characteristic functional of generalized shot noise is developed. Through expansions in terms of functional derivatives this yields expressions for moment functions of all orders. A central limit theorem also follows. Several examples are discussed.

111 citations


Journal ArticleDOI
TL;DR: In this article, the authors relax the required degree of stationarity, requiring only that the squared variables properly form a uniformly integrable family, and the partial sums have variances consistent with the Wiener process.
Abstract: In an earlier paper, the author proves an invariance principle for mixingales, a generalization of the concepts of mixing sequences and martingale differences, under the condition that the variance of the sum of $n$ random variables is asymptotic to $\sigma^2n$ where $\sigma^2 > 0$. In this note we relax further the required degree of stationarity, requiring only that the squared variables properly normalized form a uniformly integrable family, and the partial sums have variances consistent with the Wiener process.

109 citations


Journal ArticleDOI
TL;DR: In this paper, the distribution of the maxima, first entrance times, and expected occupation time densities for the two limit processes of the conditional functional central limit theorems are computed using weak convergence.
Abstract: : Brownian meander and Brownian excursion processes arise as the limit process of a number of conditional functional central limit theorems. To reap the full benefit of such limit theorems one needs to know the distribution of functionals of the limit process. In this paper the distribution of the maxima, first entrance times, and expected occupation time densities are computed for the two limit processes. This is done by using the methods of weak convergence.

105 citations


Journal ArticleDOI
TL;DR: In this paper, the strength distribution for an arbitrary excitation is given in terms of a double expansion, and its sum rules by single expansions, in polynomials defined by the initial and final energy spectra.

72 citations


Journal ArticleDOI
TL;DR: The problem of establishing a central limit theorem for the coefficients of a sequence of polynomials Pn(x) of binomial type is introduced and a complete answer is given in the case when g(u) is a polynomial.

65 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider discrete-parameter stochastic processes that are the output of a nonlinear filter driven by white noise, and derive estimates of the unknown coefficients in the transfer function and the noise variance, and investigate their asymptotic properties.

62 citations



Book ChapterDOI
TL;DR: In this article, the central limit and iterated logarithm results for B n (S n −S 221E;) where the multipliers B n ↑ ∞ a.s.
Abstract: Let {S n , n ≧ 1} be a zero, mean square integrable martingale for which \(\lim _{n \to \infty } ES_n^2 < \infty \) so that S n →S 221E; a.s., say, by the martingale convergence theorem. The paper is principally concerned with obtaining central limit and iterated logarithm results for B n (S n −S 221E;) where the multipliers B n ↑ ∞ a.s. An example on the P olya urn scheme is given to illustrate the results.

Journal ArticleDOI
TL;DR: In this article, for sums of independent and identically distributed random vectors, the authors investigated the convergence to normality of the expectation of bounded and certain unbounded Borel measurable functions, and nonuniform convergence rates.
Abstract: Speeds of convergence to normality for sums of independent and identically distributed random vectors in $\mathbb{R}^k, k \geqq 1$, are investigated using the method of operators. Results obtained improve and extend existing results on speeds of convergence for the expectations of both bounded and certain unbounded Borel measurable functions, and nonuniform convergence rates.

Journal ArticleDOI
TL;DR: In this paper, Kolmogorov's version of the law of the iterated logarithm (LIL) for random variables taking values in a real separable Banach space is established.
Abstract: : Kolmogorov's version of the law of the iterated logarithm (LIL) for random variables taking values in a real separable Banach space is established. This result has several corollaries dealing with the LIL for independent, identically distributed (i.i.d.) sequences and, in particular, extends some of the recent work of G. Pisier.

Journal ArticleDOI
Holger Rootzén1
TL;DR: In this article, necessary and sufficient conditions for convergence in distribution to a Brownian motion are obtained when the normalization is given by the sums of squares of the variables, by the conditional variances and by the variances.
Abstract: Necessary and sufficient conditions for the functional central limit theorem for a double array of random variables are sought. It is argued that this is a martingale problem only if the variables truncated at some fixed point c are asymptotically a martingale difference array. Under this hypothesis, necessary and sufficient conditions for convergence in distribution to a Brownian motion are obtained when the normalization is given (i) by the sums of squares of the variables, (ii) by the conditional variances and (iii) by the variances. The results are proved by comparing the various normalizations with a “natural” normalization.

Journal ArticleDOI
TL;DR: A survey of recent developments in the field of rates of convergence and asymptotic expansions in the context of the multi-dimensional central limit theorem can be found in this paper.
Abstract: This is an expository survey of recent developments in the field of rates of convergence and asymptotic expansions in the context of the multi-dimensional central limit theorem. A number of applications are discussed. One of them deals with normal approximations and expansions of distribution functions of a class of statistics which includes functions of sample moments.

Journal ArticleDOI
TL;DR: One-sided iterated logarithm laws of the form (1/b_n) \sum^n_1 X_i = 1, a.s. are obtained for asymmetric independent and identically distributed random variables, the first when these have a vanishing but barely finite mean, the second when $E|X|$ is barely infinite.
Abstract: One-sided iterated logarithm laws of the form $\lim \sup (1/b_n) \sum^n_1 X_i = 1$, a.s. and $\lim \sup (1/b_n) \sum^n_1 X_i = -1$, a.s. are obtained for asymmetric independent and identically distributed random variables, the first when these have a vanishing but barely finite mean, the second when $E|X|$ is barely infinite. In both cases, $\lim \inf (1/b_n) \sum^n_1 X_i = -\infty$, a.s. The constants $b_n/n$ are slowly varying, decreasing to zero in the first case and increasing to infinity in the second. Although defined via the distribution of $|X|, b_n$ represents the order of magnitude of $E|\sum^n_1 X_i|$ when this is finite. Corresponding weak laws of large numbers are established and related to Feller's notion of "unfavorable fair games" and in the process a theorem playing the same role for the weak law as Feller's generalization of the strong law is proved.

Journal ArticleDOI
TL;DR: In this article, the central limit theorem for the random element √A(PA(t)−P(t)) was proved for the Gaussian process model, and it was shown that under appropriate conditions the sequence of PA(t)-p(t ) converges to the solution of the nonlinear Volterra type integral equation.

Journal ArticleDOI
TL;DR: In this paper, the authors extended the central limit theorem for the Ising model to the statistically significant case of vertical and horizontal interactions and showed that the distortion at critical points is relatively minor.
Abstract: In Pickard (1976) limit theorems were obtained for the classical Ising model at non-critical points. These determined the asymptotic distribution of the sample nearest-neighbour correlation, thereby providing a basis for statistical inference by confidence intervals. In this paper, these limit theorems are extended to the statistically significant case of different vertical and horizontal interactions. Results at critical points are also obtained. Critical points clearly have the potential to seriously distort statistical inferences, especially in their immediate neighbourhoods. For our Ising model it turns out that such distortion is relatively minor. Surprisingly, in the two-parameter case the correlation between the sufficient statistics exhibits peculiar asymptotic behaviour resulting in a singular covariance matrix at critical points in the central limit theorem. Finally, at critical points, unusual norming constants are required for the central limit theorem, and our results are much more sensitive to the relative rate at which m , n tend to infinity.

Journal ArticleDOI
TL;DR: In this article, it was shown that the sample paths of a Robbins-Monro process with harmonic coefficients can be approximated by weighted sums of independent, identically distributed random variables.
Abstract: It is shown in this paper that the sample paths of a Robbins-Monro process with harmonic coefficients may be approximated by weighted sums of independent, identically distributed random variables. A law of iterated logarithm and a weak invariance principle follow from this result.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the set of all forests consisting of rooted trees and containing nonroot vertices; the root vertices are numbered from 1 to, and the nonroot from 1-to.
Abstract: The author considers the set of all forests consisting of rooted trees and containing nonroot vertices; the root vertices are numbered from 1 to , and the nonroot from 1 to . A uniform probability distribution is introduced on this set. Let denote a random variable equal to the number of trees of a random forest containing exactly nonroot vertices. Results are obtained yielding a complete description of the limit behavior of the variables for all values of for various ways of letting and approach infinity. It is shown that these results can be used for studying random mappings.Bibliography: 9 titles.

Journal ArticleDOI
TL;DR: In this paper, a non-commutative counterpart of weak convergence for canonical Wiener processes is presented. But it is not shown that Pn, Qn converges weakly to a canonical WIPT.

Journal ArticleDOI
Holger Rootzén1
TL;DR: In this paper, double arrays of random variables obtained by normalizing a sequence that is asymptotically close to a martingale difference sequence are considered, and conditions ensuring that the row sums converge in distribution to a mixture of normal distributions are found.
Abstract: Double arrays of random variables obtained by normalizing a sequence that is asymptotically close to a martingale difference sequence are considered, and conditions ensuring that the row sums converge in distribution to a mixture of normal distributions are found. The main condition is that the sums of squares in each row converge in probability to a random variable.

Book ChapterDOI
W. R. van Zwet1
01 Jan 1977
TL;DR: Asymptotic expansions for sums of independent and identically distributed random variables have been studied in the literature for many years as mentioned in this paper, and the main techniques for extending this theory to more general statistics are discussed in this paper.
Abstract: Publisher Summary This chapter discusses asymptotic expansions and explains their need. It reviews the classical theory of Edgeworth expansions for sums of independent and identically distributed random variables, and indicates the two main techniques for extending this theory to more general statistics. The chapter presents an account of as yet unpublished results of Bjerve and Helmers who establish Berry–Esseen type bounds for linear combinations of order statistics. For many years, mathematical statisticians have spent a great deal of effort and ingenuity toward applying the central limit theorem in statistics. The estimators and test statistics that interest statisticians are as a rule not sums of independent random variables, and much work went into showing that they can often be approximated sufficiently well by such sums to ensure asymptotic normality. This work can be traced throughout the development of mathematical statistics from the proof of the asymptotic normality of the maximum likelihood estimator to much of the recent work in nonparametric statistics.

Journal ArticleDOI
TL;DR: In this paper, it is argued that the conditions required for nonnormal stable limits are frequently satisfied by economic variables and in particular by disturbance terms in regressions, and it might be wise for economic statisticians to devote more attention to robust statistical methods.
Abstract: The heavy reliance on the normal distribution in the teaching of economic statistics is usually justified by appealing to the central limit theorem. Limit theorem arguments can, however, also lead to the nonnormal stable distributions. It is argued that the conditions required for nonnormal stable limits are, in fact, frequently satisfied by economic variables and in particular by disturbance terms in regressions. Normal theory results should, therefore, be used with caution, and it might be wise for economic statisticians to devote more attention to robust statistical methods.

Book ChapterDOI
01 Jan 1977
TL;DR: In this paper, a systematic theory of limit distributions for sums of "strongly dependent" random variables is presented, and it is shown that the central limit theorem is not valid as a working definition of strong dependence.
Abstract: The aim of these lectures is to give an introduction to some recent developments in probability theory which, besides having an independent interest, seem to provide a very effective tool in view of a mathematically rigorous description of critical phenomena. What is involved is the construction of a systematic theory of limit distributions for sums of “strongly dependent” random variables. The notion of strong dependence will be made precise later. For the moment, as a working definition, we take it to mean that the central limit theorem is not valid.



Journal ArticleDOI
TL;DR: In this article, the problem of estimating the deviation of the distribution of the sum of a random number of differently distributed random variables from the normal distribution is considered, and the Esseen inequality for distributions of random sums is established.
Abstract: The problem of estimation of deviation of the distribution of the sum of a random number of differently distributed random variables from the normal distribution is considered, i.e., the Esseen inequality for distributions of random sums is established. As a particular case, the Berry-Esseen inequality is obtained.