Showing papers on "Central limit theorem published in 1989"
Book•
01 Jan 1989
TL;DR: In this article, a review of results of the General Theory of Stochastic Processes is presented, with a focus on the weak convergence of distributions of semimartingales to distributions of point processes.
Abstract: I.- 1. Basic Concepts and the Review of Results of "The General Theory of Stochastic Processes".- 1. Stochastic basis. Random times, sets and processes.- 2. Optional and predictable ?-algebras of random sets.- 3. Predictable and totally inaccessible random times. Classification of Markov times. Section theorems.- 4. Martingales and local martingales.- 5. Square integrable martingales.- 6. Increasing processes. Compensators (dual predictable projections). The Doob-Meyer decomposition.- 7. The structure of local martingales.- 8. Quadratic characteristic and quadratic variation.- 9. Inequalities for local martingales.- 2. Semimartingales. I. Stochastic Integral.- 1. Semimartingales and quasimartingales.- 2. Stochastic integral with respect to a local martingale and a semimartingale. Construction and properties.- 3. Ito's formula. I.- 4. Doleans equation. Stochastic exponential.- 5. Multiplicative decomposition of positive semimartingales.- 6. Convergence sets and the strong law of large numbers for special martingales.- 3. Random Measures and their Compensators.- 1. Optional and predictable random measures.- 2. Compensators of random measures. Conditional mathematical expectation with respect to the ?-algebra P?.- 3. Integer-valued random measures.- 4. Multivariate point processes.- 5. Stochastic integral with respect to a martingale measure ?-?.- 6. Ito's formula. II.- 4. Semimartingales. II Canonical Representation.- 1. Canonical representation. Triplet of predictable characteristics of a semimartingale.- 2. Stochastic exponential constructed by the triplet of a semimartingale.- 3. Martingale characterization of semimartingales by means of stochastic exponentials.- 4. Characterization of semimartingales with conditionally independent increments.- 5. Semimartingales and change of probability measures. Transformation of triplets.- 6. Semimartingales and reduction of a flow of ?-algebras.- 7. Semimartingales and random change of time.- 8. Semimartingales and integral representation of martingales.- 9. Gaussian martingales and semimartingales.- 10. Filtration of special semimartingales.- 11. Semimartingales and helices. Ergodic theorems.- 12. Semimartingales - stationary processes.- 13. Exponential inequalities for large deviation probabilities.- II.- 5. Weak Convergence of Finite-Dimensional Distributions of Semimartingales to Distributions of Processes with Conditionally Independent Increments.- 1. Method of stochastic exponentials. I. Convergence of conditional characteristic functions.- 2. Method of stochastic exponentials. II. Weak convergence of finite dimensional distributions.- 3. Weak convergence of finite dimensional distributions of point processes and semimartingales to distributions of point processes.- 4. Weak convergence of finite dimensional distributions of semimartingales to distributions of a left quasi-continuous semimartingale with conditionally independent increments.- 5. The central limit theorem. I. "Classical" version.- 6. The central limit theorem. II. "Nonclassical" version.- 7. Evaluation of a convergence rate for marginal distributions in the central limit theorem.- 8. A martingale method of proving the central limit theorem for strictly stationary sequences. Relation to mixing conditions.- 6. The Space D. Relative Compactness of Probability Distributions of Semimartingales.- 1. The space D. Skorohod's topology.- 2. Continuous functions on R+ x D.- 3. Conditions on adapted processes sufficient for relative compactness of families of their distributions.- 4. Relative compactness of probability distributions of semimartingales.- 5. Conditions necessary for the weak convergence of probability distributions of semimartingales.- 7. Weak Convergence of Distributions of Semimartingales to Distributions of Processes with Conditionally Independent Increments.- 1. The functional central limit theorem (invariance principle).- 2. Weak convergence of distributions of semimartingales to distributions of point processes.- 3. Weak convergence of distributions of semimartingales to the distribution of a left quasi-continuous semimartingale, with conditionally independent increments.- 8. Weak Convergence of Distributions of Semimartingales to the Distribution of a Semimartingale.- 1. Convergence of stochastic exponentials and weak convergence of distributions of semimartingales.- 2. Weak convergence to the distribution of a left quasi-continuous semimartingale.- 3. Diffusion approximation.- 4. Weak convergence to a distribution of a point process with a continuous compensator.- 5. Weak convergence of in variant measures.- III.- 9. Invariance Principle and Diffusion Approximation for Models Generated by Stationary Processes.- 1. Generalization of Donsker's invariance principle.- 2. Invariance principle for strictly stationary processes.- 3. Invariance principle for a Markov process.- 4. Diffusion approximation for systems with a "broad bandwidth noise" (scalar case).- 5. Diffusion approximation with a "broad bandwidth noise" (vector case).- 6. Ergodic theorem and invariant principle in case of nonhomogeneous time averaging.- 7. Stochastic version of Bogoljubov's averaging principle.- 10. Diffusion Approximation for Semimartingales with a Normal Reflexion in a Convex Region.- 1. Skorohod's problem on normal reflection.- 2. Semimartingale with normal reflection.- 3. Diffusion approximation with normal reflection.- 4. Diffusion approximation with reflection for queueing models with autonomious service.- Historic-Bibliographical notes.
871 citations
TL;DR: In this article, the authors present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments.
Abstract: We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.
361 citations
TL;DR: Theoretical properties of Gaussian fields have been studied in this article, where the authors define a theory of random fields based on the central limit theorem for non-linear transformations of the Gaussian field.
Abstract: 1. Elements of the Theory of Random Fields.- 1.1 Basic concepts and notation.- 1.2 Homogeneous and isotropic random fields.- 1.3 Spectral properties of higher order moments of random fields.- 1.4 Some properties of the uniform distribution.- 1.5 Variances of integrals of random fields.- 1.6 Weak dependence conditions for random fields.- 1.7 A central limit theorem.- 1.8 Moment inequalities.- 1.9 Invariance principle.- 2. Limit Theorems for Functionals of Gaussian Fields.- 2.1 Variances of integrals of local Gaussian functionals.- 2.2 Reduction conditions for strongly dependent random fields.- 2.3 Central limit theorem for non-linear transformations of Gaussian fields.- 2.4 Approximation for distribution of geometric functional of Gaussian fields.- 2.5 Reduction conditions for weighted functionals.- 2.6 Reduction conditions for functionals depending on a parameter.- 2.7 Reduction conditions for measures of excess over a moving level.- 2.8 Reduction conditions for characteristics of the excess over a radial surface.- 2.9 Multiple stochastic integrals.- 2.10 Conditions for attraction of functionals of homogeneous isotropic Gaussian fields to semi-stable processes.- 3. Estimation of Mathematical Expectation.- 3.1 Asymptotic properties of the least squares estimators for linear regression coefficients.- 3.2 Consistency of the least squares estimate under non-linear parametrization.- 3.3 Asymptotic expansion of least squares estimators.- 3.4 Asymptotic normality and convergence of moments for least squares estimators.- 3.5 Consistency of the least moduli estimators.- 3.6 Asymptotic normality of the least moduli estimators.- 4. Estimation of the Correlation Function.- 4.1 Definition of estimators.- 4.2 Consistency.- 4.3 Asymptotic normality.- 4.4 Asymptotic normality. The case of a homogeneous isotropic field.- 4.5 Estimation by means of several independent sample functions.- 4.6 Confidence intervals.- References.- Comments.
250 citations
TL;DR: In this article, a simple martingale argument is presented which proves that directed polymers in random environments satisfy a central limit theorem ford ≥ 3 and if the disorder is small enough.
Abstract: A simple martingale argument is presented which proves that directed polymers in random environments satisfy a central limit theorem ford≧3 and if the disorder is small enough. This simplifies and extends an approach by J. Imbrie and T. Spencer.
210 citations
TL;DR: Stein's method of obtaining rates of convergence to the normal distribution is illustrated in the context of random graph theory and results are obtained for the number of copies of a given graph G in K.
173 citations
TL;DR: In this article, the central limit of the CCR-C*-algebra of fluctuations is studied by means of the notion of relative entropy, and non-commutative central limit theorems are derived.
Abstract: Non-commutative central limit theorems are derived. The CCR-C
*-algebra of fluctuations is analyzed in detail. The stability of the central limit is studied by means of the notion of relative entropy.
130 citations
114 citations
TL;DR: In this article, it was shown that a recurrent random walk with centered random variables satisfies a central limit theorem, and a functional version is presented. But the central limit is not satisfied in the case where the variables are centered.
Abstract: Let $S_n, n \in \mathbb{N}$, be a recurrent random walk on $\mathbb{Z}^2 (S_0 = 0)$ and $\xi(\alpha), \alpha \in \mathbb{Z}^2$, be i.i.d. $\mathbb{R}$-valued centered random variables. It is shown that $\sum^n_{i = 1}\xi(S_i)/ \sqrt{n \log n}$ satisfies a central limit theorem. A functional version is presented.
110 citations
TL;DR: In this paper, the central limit theorem for real-valued random variables has been shown to be monotone in the sense of Zarantarello's theorem in Hilbert spaces.
Abstract: Let $(X, Y)$ be a pair of Hilbert-valued random variables for which the Wasserstein distance between the marginal distributions is reached We prove that the mapping $\omega \rightarrow (X(\omega), Y(\omega))$ is increasing in a certain sense Moreover, if $Y$ satisfies a nondegeneration condition, we can take $X = T(Y)$ with $T$ monotone in the sense of Zarantarello We apply these results to obtain a proof of the central limit theorem (CLT) in Hilbert spaces which does not make use of the CLT for real-valued random variables
89 citations
TL;DR: It is shown, under mild regularity conditions, that any nonlinear control- variable scheme is equivalent to a linear control-variable scheme from the point of view of asymptotic efficiency, and that the issue of indirect-versus-direct estimation is related to estimation using non linear control variables.
Abstract: For a large class of queueing systems, Little's law (L = λW) helps provide a variety of statistical estimators for the long-run time-average queue length L and the long-run customer-average waiting time W. We apply central limit theorem versions of Little's law to investigate the asymptotic efficiency of these estimators. We show that an indirect estimator for L using the natural estimator for W plus the known arrival rate λ is more efficient than a direct estimator for L, provided that the interarrival and waiting times are negatively correlated, thus extending a variance-reduction principle for the GI/G/s model due to A. M. Law and J. S. Carson. We also introduce a general framework for indirect estimation which can be applied to other problems besides L = λW. We show that the issue of indirect-versus-direct estimation is related to estimation using nonlinear control variables. We also show, under mild regularity conditions, that any nonlinear control-variable scheme is equivalent to a linear control-va...
86 citations
01 Jan 1989
TL;DR: In this paper, the authors discuss the conditions for the approximate normality of the distribution of the number of local maxima of a random function on the set of vertices of a graph when the values of the random function are independently identically distributed with a continuous distribution function.
Abstract: Publisher Summary This chapter discusses the normal approximation for the number of local maxima of a random function on a graph. It discusses the conditions for the approximate normality of the distribution of the number of local maxima of a random function on the set of vertices of a graph when the values of the random function are independently identically distributed with a continuous distribution function. For a regular graph, the distribution of the number of local maxima is approximately normal if its variance is large. The basic idea of a normal approximation theorem is to exploit a sum of indicator random variables. The chapter discusses a basic lemma on normal approximation for sums of indicator random variables.
TL;DR: It was proved in this article that $EX^2 < \infty$ is necessary for a very mild form of the bootstrap of the mean to work as and that $X$ must be in the domain of attraction of the normal law if as is weakened to "in probability".
Abstract: It is proved that $EX^2 < \infty$ is necessary for a very mild form of the bootstrap of the mean to work as and that $X$ must be in the domain of attraction of the normal law if as is weakened to "in probability"
TL;DR: In this article, the authors studied the convergence rate of the central limit theorem for functions of independent random variables and applied it to the problem of estimating the probability of a function of independent variables.
Abstract: The rate of convergence in the central limit theorem for functions of independent random variables is studied in a unifying approach. The basic result sharpens and extends a theorem of van Zwet. Applications to $U$-, $L$- and $R$-statistics are also given, improving or extending the results of Helmers and van Zwet, Helmers and Huskova, Does and van Es and Helmers.
01 Nov 1989
TL;DR: In this article, a Lyapunov function criterion was developed to bound the solution g to Poisson's equation for a positive recurrent Harris chain with invariant measure pi, and then applied to obtain sufficient conditions that guarantee that the solution be an element of L sub p (pi) when p = 2.
Abstract: : Suppose that X is a positive recurrent Harris chain with invariant measure pi We develop a Lyapunov function criterion that permits one to bound the solution g to Poisson's equation for X This bound is then applied to obtain sufficient conditions that guarantee that the solution be an element of L sub p (pi) When p = 2, the square integrability of g implies the validity of a functional central limit theorem for the Markov chain We illustrate the technique with applications to the waiting time sequence of the single-server queue and autoregressive sequences Keywords: Functional central limit theorem
TL;DR: In this article, the authors investigated the behavior of moment determinants of certain moment matrices, for which the entry is the $(i, j)$th moment of a distribution.
Abstract: An investigation is carried out in the behavior of the determinants of certain moment matrices, for which the $(i, j)$ entry is the $(i + j)$th moment of a distribution $F$. The determinant can be represented as the expected value of a $U$-statistic type kernel. The structure of the kernel illustrates how the determinant carries information about the number of support points of the distribution $F$. The kernel representation can be extended to the determinant of a matrix of moment generating function derivatives, where the $(i, j)$ entry is the $i + j$th derivative of the moment generating function of $F$. When done, this reveals that this determinant is itself, as a function of $t$, a moment generating function. When this somewhat surprising result is applied to members of the quadratic variance exponential family, one obtains the result that they are closed under this two-step operation of taking derivatives, then computing determinants. This results in an elementary recursion for the values of the moment determinants. The final result gives the convergence of the moment determinants to the normal theory values under central limit theorem conditions.
TL;DR: In this paper, it was shown that for all neN and all xeR where c1 < 31.935, the analytical structure of C1=C1 is given without numerical calculation.
Abstract: Let X1,X2 be a sequence of independent random variable such that . It is shown that for all neN and all xeR where c1<31.935 In the general case only tha analytical structure of C1=C1 is given without numerical calculation
Posted Content•
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.
Abstract: This paper establishes a central limit theorem (CLT) for empirical processes indexed by smooth functions. The underlying random variables may be temporally dependent and non-identically distributed. In particular, the CLT holds for near epoch dependent (i.e., functions of mixing processes) triangular arrays, which include strong mixing arrays, among others. The results apply to classes of functions that have series expansions. The proof of the CLT is particularly simple; no chaining argument is required. The results can be used to establish the asymptotic normality of semiparametric estimators in time series contexts. An example is provided.
TL;DR: In this paper, a control chart for process centrality and dispersion with inverse Gaussian distributed output is presented for processes with 10 or fewer items and the central limit theorem is invoked.
Abstract: summary
Control charts typically used to monitor process centrality and dispersion either assume normally distributed output or appeal to the viability of the central limit theorem. This appeal to the central limit theorem is frequently made on behalf of samples consisting of 10 or fewer items. Control charts for centrality and dispersion are presented for processes with inverse Gaussian distributed output.
TL;DR: In this paper, a strictly stationary finite-state non-degenerate random sequence is constructed which satisfies pairwise independence and absolute regularity but fails to satisfy a central limit theorem.
Abstract: A strictly stationary finite-state non-degenerate random sequence is constructed which satisfies pairwise independence and absolute regularity but fails to satisfy a central limit theorem. The mixing rate for absolute regularity is only slightly slower than that in a corresponding central limit theorem of Ibragimov.
TL;DR: In this paper, the authors consider the set of mappings of the integers (1, 2, \ldots, n\} into a uniform probability measure on this set and study the component structure of the associated graphs as $n \rightarrow \infty.
Abstract: We consider the set of mappings of the integers $\{1, 2, \ldots, n\}$ into $\{1, 2, \ldots, n\}$ and put a uniform probability measure on this set. Any such mapping can be represented as a directed graph on $n$ labelled vertices. We study the component structure of the associated graphs as $n \rightarrow \infty$. To each mapping we associate a step function on $\lbrack 0, 1 \rbrack$. Each jump in the function equals the number of connected components of a certain size in the graph which represents the map. We normalize these functions and show that the induced measures on $D\lbrack 0, 1 \rbrack$ converge to Wiener measure. This result complements another result by Aldous on random mappings.
TL;DR: In this paper, an asymptotic expansion for the distribution of the sum of independent zero-one random variables in case where this surname has variance σ n 2 → ∞ is given.
TL;DR: It is verified that when the sample size is large enough the application of the Central Limit Theorem determines an operative extension of the Neyman-Pearson optimality criterion.
TL;DR: In this paper, the authors develop a method to get empirical central limit theorems for martingale differences that are uniformly bounded, based on the ideas of E. Gine and J. Zinn.
Abstract: This paper develops a method to get empirical central limit theorems for martingale differences that are uniformly bounded. The main idea is to generalize to martingales some ideas of E. Gine and J. Zinn [Ann. Prob.12, 929–989 (1984)]. We consider two examples: An extension of a theorem of R. Dudley from i.i.d. to a certain type of Markov chain, and a uniform CLT for the “baker's transformation”.
01 Jan 1989
TL;DR: In this article, the Parthasarathy-Meyer method was extended to deal with arbitrary d-dimensional Bernoulli processes, where $d$ is a natural integer (cfr.
Abstract: In [1] we have proved a quantum De Moivre-Laplace theorem based on a
modification of the Giri-von Waldenfels quantum central limit theorem. In [2] P.A. Meyer outlined a method based on direct calculations which, taking advantage of the explicit structure of the algebra of $2\times 2$ matrices, allows a drastic simplification of the proof of the main result of the first part of our paper and relates it with a similar result obtained, independently and simultaneusly, by Parthasarathy [5].
In the first part of the present note we simplify the Parthasarathy-Meyer method and extend it to deal with arbitrary d-dimensional Bernoulli
processes, where $d$ is a natural integer (cfr. Sections (3),(4).
We also prove another statement in Meyer's note (cf. Theorem (5.1)).
Finally (Section (6)) we show that the method of proof used in [1] allows, with minor modifications, to solve the problem of the central limit approximation of the squeezing states - a problem left open in [1] and to which, due to the nonlinearity of the coupling, Parthasarathy-Meyer
direct computational method cannot be applied.
05 Jun 1989
TL;DR: Simulations confirm that Brownian motion with drift is an accurate model of system performance and the distribution of finishing time is very tight about its mean and well approximated by a normal distribution.
Abstract: Distributed systems having large numbers of idle computers and workstations are analyzed using a very simple model of a distributed program (a fixed amount of work) to see how the use of transient processors affects the program's service time. The probability density of the length of time it takes to finish a fixed amount of work is determined. An equation is given for the main result for an M-processor network. Simulations confirm that Brownian motion with drift is an accurate model of system performance. With large programs that run for a long time relative to the length of available and nonavailable periods, the central limit-theorem applies, and the Brownian-motion-with-drift model remains good regardless of the distributions of the available and the nonavailable periods. Under these assumptions, the distribution of finishing time is very tight about its mean and well approximated by a normal distribution. >
13 Dec 1989
TL;DR: In this article, the central limit theorem holds for discrete-event dynamic systems and the cycle time of the system is asymptotically normally distributed, and the expectation and variance of cycle time are given.
Abstract: Various aspects of the asymptotic behavior of discrete-events dynamic systems (DEDS) in which the activity times are random variables are discussed. The main result is that the central limit theorem holds for DEDS and consequently that the cycle time of the system is asymptotically normally distributed. Calculations of the expectation and variance of the cycle time are given. Reducible random DEDS are considered, and the behavior of random DEDS is compared with that of deterministic DEDS. >
TL;DR: Expansions for the distribution of differentiable functionals of normalized sums of i.i.d. random vectors taking values in a separable Banach space are derived in this article.
Abstract: Expansions for the distribution of differentiable functionals of normalized sums of i.i.d. random vectors taking values in a separable Banach space are derived. Assuming that an $(r + 2)$th absolute moment exist, the CLT holds and the distribution of the $r$th derivative $r \geq 2$ of the functionals under the limiting Gaussian law admits a Lebesgue density which is sufficiently many times differentiable, expansions up to an order $O(n^{-r/2 + \varepsilon})$ hold. Applications to goodness-of-fit statistics, likelihood ratio statistics for discrete distribution families, bootstrapped confidence regions and functionals of the uniform empirical process are investigated.
TL;DR: In this article, the authors modify the likelihood-based method for obtaining derivatives with respect to the rate of a Poisson process to that it is not necessary to know the exact value of that rate.
Abstract: We modify the likelihood-based method for obtaining derivatives with respect to the rate of a Poisson process to that it is not necessary to know the exact value of that rate. This type of modification is necessary if the method is to be used on a sample path from a real system. The method is also applicable to simulation studies of certain real time control policies and may be useful in trace driven simulations. The modification to the likelihood estimator is simply to use the value of the Poisson rate estimated during the sample interval. For regenerative systems, this produces a strongly consistent, asymptotically normal and asymptotically unbiased estimate of the derivative. The strong law and central limit theorem are generalized to the case of estimating a derivative with respect to an unknown parameter from the exponential class of probability density functions. Numerical results for the M/M/1 queue illustrate little difference between the estimates for the derivative of the expected delay with respect to arrival rate obtained when the arrival rate is known and unknown. However, both estimates are highly biased for small sample sizes. This bias can be reduced by jackknifing.
TL;DR: A central limit theorem for orthogonally invariant random variables on Pn, the space of n × n real, positive definite symmetric matrices, was obtained in this paper.