Showing papers on "Central limit theorem published in 1991"
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01 May 1991
TL;DR: Banach Space Valued Random Variables and their strong limiting properties are discussed in this paper, where the central limit theorem and the strong law of large numbers are also considered, as well as the regularity of Gaussian and Stable Processes.
Abstract: Notation.- 0. Isoperimetric Background and Generalities.- 1. Isoperimetric Inequalities and the Concentration of Measure Phenomenon.- 2. Generalities on Banach Space Valued Random Variables and Random Processes.- I. Banach Space Valued Random Variables and Their Strong Limiting Properties.- 3. Gaussian Random Variables.- 4. Rademacher Averages.- 5. Stable Random Variables.- 6 Sums of Independent Random Variables.- 7. The Strong Law of Large Numbers.- 8. The Law of the Iterated Logarithm.- II. Tightness of Vector Valued Random Variables and Regularity of Random Processes.- 9. Type and Cotype of Banach Spaces.- 10. The Central Limit Theorem.- 11. Regularity of Random Processes.- 12. Regularity of Gaussian and Stable Processes.- 13. Stationary Processes and Random Fourier Series.- 14. Empirical Process Methods in Probability in Banach Spaces.- 15. Applications to Banach Space Theory.- References.
1,926 citations
TL;DR: In this paper, it was shown that free random variables naturally arise as limits of random matrices and that Wigner's semicircle law is a consequence of the central limit theorem for free variables.
Abstract: In earlier articles we studied a kind of probability theory in the framework of operator algebras, with the tensor product replaced by the free product. We prove here that free random variables naturally arise as limits of random matrices and that Wigner's semicircle law is a consequence of the central limit theorem for free random variables. In this way we obtain a non-commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra, Wigner's law being given by the trace of the spectral measure of the selfadjoint component of this operator
891 citations
TL;DR: In this paper, the error in the central limit theorem for sums and convex sets is estimated using Stein's method and induction. But the error for multivariate functions of independent random elements is not considered.
Abstract: Berry-Esseen theorems are proved in the multidimensional central limit theorem without using Fourier methods. An effective and simple estimate of the error in the CLT for sums and convex sets using Stein's method and induction is derived. Furthermore, the error in the CLT for multivariate functions of independent random elements is estimated extending results of van Zwet and Friedrich to the multivariate case.
339 citations
TL;DR: In this paper, the authors proved limit theorems for the range of d-dimensional random walks in the domain of attraction of a stable process of index β, where β is the number of distinct sites of the random walk before time n. The results depend on the value of the ratio β/d.
Abstract: Limit theorems are proved for the range of $d$-dimensional random walks in the domain of attraction of a stable process of index $\beta$. The range $R_n$ is the number of distinct sites of $\mathbb{Z}^d$ visited by the random walk before time $n$. Our results depend on the value of the ratio $\beta/d$. The most interesting results are obtained for $2/3 1$, which can only occur if $d = 1$, we prove the convergence in distribution of $R_n/E(R_n)$ toward some constant times the Lebesgue measure of the range of the limiting stable process. Some of our results require regularity assumptions on the characteristic function of $X$.
132 citations
TL;DR: In this article, a lower bound on the entropy produced when independent random variables are summed and rescaled is established. But this lower bound is based on a Lyapunov functional governing approach to the Gaussian limit.
Abstract: We prove a strict lower bound on the entropy produced when independent random variables are summed and rescaled. Using this, we develop an approach to central limit theorems from a dynamical point of view in which the entropy is a Lyapunov functional governing approach to the Gaussian limit. This dynamical approach naturally extends to cover dependent variables, and leads to new results in pure probability theory as well as in statistical mechanics. It also provides a unified framework within which many previous results are easily derived.
114 citations
TL;DR: In this article, the authors provide an expository discussion of the basic structure of the asymptotic theory of M-estimators in dynamic nonlinear models and a review of the literature.
Abstract: This is the second of two papers that provide an expository discussion of the basic structure of the asymptotic theory of M-estimators in dynamic nonlinear models and a review of the literature. The first paper, Potscher and Prucha(1991), deals with consistency. In the present paper we discuss asymptotic normality. As an important ingredient to the asymptotic normality proof in dynamic nonlinear models we consider central limit theorems for dependent random variables. We also discuss the estimation of the variance covariance matrix of m-estimators under heteroscedasticity and autocorrelation.
112 citations
TL;DR: The functional central limit theorem and law of the iterated logarithm for the Pickands estimator of the dependence function of a bivariate extreme value distribution were established in this article.
100 citations
TL;DR: In this article, the Levy stable laws are introduced to describe intermittency in multiparticle production, using a generalization of the central limit theorem applied to random cascading models, and a one-parameter family of representative models depending continuously on the Levy index μ.
68 citations
68 citations
TL;DR: In this article, the one-dimensional weakly asymmetric nearest neighbor exclusion process is considered and the fluctuations of the associated density field around the solution of the nonlinear BURGERS equation with viscosity are studied.
Abstract: We consider the one-dimensional weakly asymmetric nearest neighbour exclusion process and study, in macroscopic space-time coordinates, the fluctuations of the associated density field around the solution of the nonlinear BURGERS equation with viscosity. We show that this fluctuations converge to a generalized ORNSTEIN-UHLENBECK process, the drift term of which can be obtained by linearization of the BURGERS equation. Our approach is based on a nonlinear transformation of the exclusion process.
53 citations
TL;DR: In this article, a Hilbert space valued Markov process is compared to a deterministic model by means of a law of large numbers and central limit theorem, and nearly optimal results are obtained by exploiting the Ornstein-Uhlenbeck type structure of the stochastic model.
Abstract: Particles placed in $N$ cells on the unit interval give birth or die according to linear rates. Adjacent cells are coupled by diffusion with a rate proportional to $N^2$. Cell numbers are divided by a density parameter to represent concentrations, and the resulting space-time Markov process is compared to a corresponding deterministic model, the solution to a partial differential equation. The models are viewed as Hilbert space valued processes and compared by means of a law of large numbers and central limit theorem. New and nearly optimal results are obtained by exploiting the Ornstein-Uhlenbeck type structure of the stochastic model.
TL;DR: In this paper, the authors consider the problem of identifying the class of time series model to which a series belongs based on observation of part of the series, and study cumulative versions of Tukey regressogram estimators of such functions.
Abstract: : We consider the problem of identifying the class of time series model to which a series belongs based on observation of part of the series. Techniques of nonparametric estimation have been applied to this problem by various authors using kernel estimates of the one-step lagged conditional mean and variance functions. We study cumulative versions of Tukey regressogram estimators of such functions. These are more stable than estimates of the mean and variance functions themselves and can be used to construct confidence bands. Goodness of fit tests for specific parametric models are also developed.
TL;DR: In this article, central limit theorems for multivariate density estimators are proven for the L_p$-norms of multivariate densities, where p < ∞.
Abstract: Central limit theorems are proven for $L_p$-norms $(1 \leq p < \infty)$ of multivariate density estimators.
TL;DR: In this paper, a functional central limit theorem for Markov chains in random environments under the assumption of existence of a finite invariant, ergodic measure and a mixing condition was established.
Abstract: A functional central limit theorem is established for Markov chains in random environments under the assumption of existence of a finite invariant, ergodic measure and a mixing condition. These conditions are always satisfied when the state space is finite.
TL;DR: In this paper, the authors derived exact and limiting distributions for the size, the number of leaves and number of internal nodes of the subtree rooted at a fixed point in a random recursive tree of order n under the assumption that n is fixed and n is rightarrow \infty.
Abstract: We study the structure of $T^{(k)}_n$, the subtree rooted at $k$ in a random recursive tree of order $n$, under the assumption that $k$ is fixed and $n \rightarrow \infty$. Employing generalized Polya urn models, exact and limiting distributions are derived for the size, the number of leaves and the number of internal nodes of $T^{(k)}_n$. The exact distributions are given by intricate formulas involving Eulerian numbers, but a recursive argument based on the urn model suffices for establishing the first two moments of the above-mentioned random variables. Known results show that the limiting distribution of the size of $T^{(k)}_n$, normalized by dividing by $n$ is $\operatorname{Beta}(1, k - 1)$. A martingale central limit argument is used to show that the difference between the number of leaves and the number of internal nodes of $T^{(k)}_n$, suitably normalized, converges to a mixture of normals with a $\operatorname{Beta}(1, k - 1)$ as the mixing density. The last result allows an easy determination of limiting distributions of suitably normalized versions of the number of leaves and the number of internal nodes of $T^{(k)}_n$.
TL;DR: In this article, a functional central limit theorem and a functional law of the iterated logarithm for quadratic forms in independent random variables were proved for the case of independent random variable.
TL;DR: A direct-sequence spread-spectrum multiple-access bit-error probability analysis is developed using large-deviations theory, and it is here that these methods provide some important practical insight.
Abstract: A direct-sequence spread-spectrum multiple-access bit-error probability analysis is developed using large-deviations theory. Let m denote the number of interfering spread-spectrum signals and let n denote the signature sequence length. Then the large deviations limit is as n to infinity with m fixed. A tight asymptotic expression for the bit-error probability is proven, and in addition, recent large-deviations results with the importance sampling Monte Carlo estimation technique are applied to obtain accurate and computationally efficient estimates of the bit-error probability for finite values of m and n. The large-deviations point of view is compared also to the conventional asymptotics of central limit theory and the associated Gaussian approximation. The Gaussian approximation is accurate and the ratio m/n is moderately large and all signals have roughly equal power. In the near/far situation, however, the Gaussian approximation is quite poor. In contrast, large-deviations techniques are more accurate in the near/far situation, and it is here that these methods provide some important practical insight. >
01 Jan 1991
TL;DR: The central limit theorem as discussed by the authors is derived from ergodic theory by following a particular string of numbers or by studying the empirical distribution of the partial sums, and it can be expressed as a set of weak-convergence theorems.
Abstract: Publisher Summary This chapter presents the central limit theorem. The central limit theorem is derived from ergodic theory by following a particular string of numbers or by studying the empirical distribution of the partial sums. The central limit theorem has a number of variants. In its common form, the random variables should be identically distributed. The central limit theorem is any of a set of weak-convergence theorems. They all express the fact that a sum of many independent and identically distributed random variables, or alternatively, random variables with specific types of dependence, would be distributed according to one of a small set of attractor distributions.
TL;DR: A sufficient condition for a sequence of partial-sum set-indexed processes with non-uniform mixing condition to converge to Brownian motion is given in this paper, which is an extension of the similar results of Goldie and Greenwood by weakening the $\phi$-mixing condition.
Abstract: A sufficient condition is given for a sequence of partial-sum set-indexed processes with nonuniform $\phi$-mixing condition to converge to Brownian motion. The main result (Theorem 1.1) is an extension of the similar results of Goldie and Greenwood by weakening the $\phi$-mixing condition. An application (Corollary 4.2) to certain Gibbs fields is given.
TL;DR: In this paper, the authors established a central limit theorem for empirical processes indexed by smooth functions and used it to establish the asymptotic normality of semiparametric estimators in time series contexts.
TL;DR: In this paper, the authors apply the stochastic calculus of multiple Wiener-Ito integral expansions to express the number of crossings of the mean level by a stationary (discrete- or continuous-time) Gaussian process within a fixed time interval.
Abstract: This paper applies the stochastic calculus of multiple Wiener-Ito integral expansions to express the number of crossings of the mean level by a stationary (discrete- or continuous-time) Gaussian process within a fixed time interval [0,T]. The resulting expansions involve a class of hypergeometric functions, for which recursion and differential relations and some asymptotic properties are derived. The representation obtained for level-crossing counts is applied to prove a central limit theorem of Cuzick (1976) for level crossings in continuous time, using a general central limit theorem of Chambers and Slud (1989a) for processes expressed via multiple Wiener-Ito integral expansions in terms of a stationary Gaussian process. Analogous results are given also for discrete-time processes. This approach proves that the limiting variance is strictly positive, without additional assumptions needed by Cuzick.
TL;DR: In this paper, a connection between the Azema martingales and certain quantum stochastic processes with increments satisfying q-commutation relations was established, leading to a theory of q-white noise on q-*-bialgebras and to a generalization of the Fock space representation theorem for white noise on *-bial gebras.
Abstract: We establish a connection between the Azema martingales and certain quantum stochastic processes with increments satisfyingq-commutation relations. This leads to a theory ofq-white noise onq-*-bialgebras and to a generalization of the Fock space representation theorem for white noise on *-bialgebras. In particular, quantum Azema noise,q-interpolations between Fermion and Boson quantum Brownian motion and unitary evolutions withq-independent multiplicative increments are studied. It follows from our results that the Azema martingales and theq-interpolations are central limits of sums ofq-independent, identically distributed quantum random variables.
TL;DR: The functional central limit theorem for the Ewens sampling formula has been proved in this article, which is the first proof of functional central limits for the sampling formula in population genetics.
Abstract: The Ewens sampling formula arises in population genetics and the study of random permutations as a probability distribution on the set of partitions (by allelic type in a sample, or according to cycle structure, respectively) of the integer $n$ for each $n$. It may be embedded naturally in the familiar linear birth process with immigration. One consequence of this is another proof of the functional central limit theorem for the Ewens sampling formula.
TL;DR: In this article, the authors present the relationship between probabilistic independence and the joint cumulants (i.e., semi-invariants) of random variables, and show the convenience of applying the cumulant for the central limit theorem, the calculation of statistics for the Morison wave force, and the statistics of wave profiles using second-order stochastic Stake's...
Abstract: The assumption of probabilistic independence is commonly made in engineering analysis to gain mathematical tractability. Although the criteria of the independence of random variables have been presented in many different ways, such as in terms of probability‐distribution functions, mathematical expectations, etc., a criterion in terms of joint cumulants has not yet been properly described. This paper presents the relationship between probabilistic independence and the joint cumulants (i.e. semi‐invariants) of random variables. When a random variable is a linear or nonlinear combination of a number of independent random variables, the calculation for the statistics of this random sum is more convenient using cumulants than central moments or moments. Examples that show the convenience of applying cumulants are presented, including a simple proof for the central limit theorem, the calculation of statistics for the Morison wave force, and the statistics of wave profiles using second‐order stochastic Stake's ...
TL;DR: In this article, the Cramer $c$-condition on the joint distribution of all the components of the vector variable is defined and the Edgeworth expansion is established for functions of sample means when only one or a few components satisfy this condition.
Abstract: Many important statistics can be written as functions of sample means of vector variables. A fundamental contribution to the Edgeworth expansion for functions of sample means was made by Bhattacharya and Ghosh. In their work the crucial Cramer $c$-condition is assumed on the joint distribution of all the components of the vector variable. However, in many practical situations, only one or a few of the components satisfy (conditionally) this condition while the rest do not (such a case is referred to as satisfying the partial Cramer $c$-condition). The purpose of this paper is to establish Edgeworth expansions for functions of sample means when only the partial Cramer $c$-condition is satisfied.
TL;DR: In this paper, the authors derived a Khintchine decomposition of probability measures on symmetric hypergroups based on a triangular central limit theorem and a discussion of conditions ensuring that the set of all factors of a probability measure is weakly compact.
Abstract: Generalizing known results for special examples, we derive a Khintchine type decomposition of probability measures on symmetric hypergroups. This result is based on a triangular central limit theorem and a discussion of conditions ensuring that the set of all factors of a probability measure is weakly compact. By our main result, a probability measure satisfying certain restrictions can be written as a product of indecomposable factors and a factor in I0(K), the set of all measures having decomposable factors only. Some contributions to the classification of I0(K) are given for general symmetric hypergroups and applied to several families of examples like finite symmetric hypergroups and hypergroup joins. Furthermore, all results are discussed in detail for a class of discrete symmetric hypergroups which are generated by infinitely many joins, for a class of countable compact hypergroups, for Sturm-Liouville hypergroups on [0, ∞[ and, finally, for polynomial hypergroups.
TL;DR: In this article, a sequence of independent nonnegative random variables with common distribution function $F$ is censored on the right by another sequence of identically distributed random variables, and these two sequences are also assumed to be independent.
Abstract: A sequence of independent nonnegative random variables with common distribution function $F$ is censored on the right by another sequence of independent identically distributed random variables. These two sequences are also assumed to be independent. We estimate the density function $f$ of $F$ by a sequence of kernel estimators $f_n(t) = (\int^\infty_{-\infty}K((t - x)/h(n))d\hat{F}_n(x))/h(n),$ where $h(n)$ is a sequence of numbers, $K$ is kernel density function and $\hat{F}_n$ is the product-limit estimator of $F.$ We prove central limit theorems for $\int^T_0|f_n(t) - f(t)|^p d\mu(t), 1 \leq p < \infty, 0 < T \leq \infty,$ where $\mu$ is a measure on the Borel sets of the real line. The result is tested in Monte Carlo trials and applied for goodness of fit.
TL;DR: In this article, a broad subset of the range of the infinitesimal generator of the diffusion is identified, for functions in this set functional central limit theorems and laws of iterated logarithm are derived.
TL;DR: In this article, the dependence structures of record values and inter-record counts are studied, and a necessary and sufficient condition for the point process to have a geometric tail is given.