Showing papers on "Central limit theorem published in 1978"

Probability theory: Independence, interchangeability, martingales

Abstract: 1 Classes of Sets, Measures, and Probability Spaces.- 1.1 Sets and set operations.- 1.2 Spaces and indicators.- 1.3 Sigma-algebras, measurable spaces, and product spaces.- 1.4 Measurable transformations.- 1.5 Additive set functions, measures, and probability spaces.- 1.6 Induced measures and distribution functions.- 2 Binomial Random Variables.- 2.1 Poisson theorem, interchangeable events, and their limiting probabilities.- 2.2 Bernoulli, Borel theorems.- 2.3 Central limit theorem for binomial random variables, large deviations.- 3 Independence.- 3.1 Independence, random allocation of balls into cells.- 3.2 Borel-Cantelli theorem, characterization of independence, Kolmogorov zero-one law.- 3.3 Convergence in probability, almost certain convergence, and their equivalence for sums of independent random variables.- 3.4 Bernoulli trials.- 4 Integration in a Probability Space.- 4.1 Definition, properties of the integral, monotone convergence theorem.- 4.2 Indefinite integrals, uniform integrability, mean convergence.- 4.3 Jensen, Holder, Schwarz inequalities.- 5 Sums of Independent Random Variables.- 5.1 Three series theorem.- 5.2 Laws of large numbers.- 5.3 Stopping times, copies of stopping times, Wald's equation.- 5.4 Chung-Fuchs theorem, elementary renewal theorem, optimal stopping.- 6 Measure Extensions, Lebesgue-Stieltjes Measure,Kolmogorov Consistency Theorem.- 6.1 Measure extensions, Lebesgue-Stieltjes measure 165 6.2 Integration in a measure space.- 6.3 Product measure, Fubini's theorem, n-dimensional Lebesgue-Stieltjes measure.- 6.4 Infinite-dimensional product measure space, Kolmogorov consistency theorem.- 6.5 Absolute continuity of measures, distribution functions Radon-Nikodym theorem.- 7 Conditional Expectation, Conditional Independence, Introduction to Martingales.- 7.1 Conditional expectations.- 7.2 Conditional probabilities, conditional probability measures.- 7.3 Conditional independence, interchangeable random variables.- 7.4 Introduction to martingales.- 7.5 U-statistics.- 8 Distribution Functions and Characteristic Functions.- 8.1 Convergence of distribution functions, uniform integrability, Helly-Bray theorem.- 8.2 Weak compactness, Frechet-Shohat, GlivenkoCantelli theorems.- 8.3 Characteristic functions, inversion formula, Levy continuity theorem.- 8.4 The nature of characteristic functions, analytic characteristic functions, Cramer-Levy theorem.- 8.5 Remarks on k-dimensional distribution functions and characteristic functions.- 9 Central Limit Theorems.- 9.1 Independent components.- 9.2 Interchangeable components.- 9.3 The martingale case.- 9.4 Miscellaneous central limit theorems.- 9.5 Central limit theorems for double arrays.- 10 Limit Theorems for Independent Random Variables.- 10.1 Laws of large numbers.- 10.2 Law of the iterated logarithm.- 10.3 Marcinkiewicz-Zygmund inequality, dominated ergodic theorems.- 10.4 Maxima of random walks.- 11 Martingales.- 11.1 Uperossing inequality and convergence.- 11.2 Martingale extension of Marcinkiewicz-Zygmund inequalities.- 11.3 Convex function inequalities for martingales.- 11.4 Stochastic inequalities.- 12 Infinitely Divisible Laws.- 12.1 Infinitely divisible characteristic functions.- 12.2 Infinitely divisible laws as limits.- 12.3 Stable laws.

Surface topography as a nonstationary random process

Abstract: TOPOGRAPHY is often considered as a narrow bandwidth of features covering the form or shape of the surface. After detailed study of many measurements we consider that as well as the possibility of a dominant range of features there is always an underlying random structure where undulations in surface height continue over as broad a bandwidth as the surface size will allow. We consider this a result of many physical effects each confined to a specific waveband but no band being dominant. We invoke the central limit theorem and show through Gaussian statistics that the variance of the height distribution of such a structure is linearly related to the length of sample involved. In another form, the power spectral density, this relationship is shown to agree well with measurements of structures taken over many scales of size, and from throughout the physical universe.

On Mixing and Stability of Limit Theorems

Abstract: convergence of random variables. In this expository note we point out some equivalent definitions of mixing and stability and discuss the use of these concepts in several contexts. Further, we show how a recent central limit theorem for martingales can be obtained directly using stability. Though the results are not new, the proofs seem substantially simpler than those previously given.

The Strong Limits of Random Matrix Spectra for Sample Matrices of Independent Elements

Abstract: This paper proves almost-sure convergence of the empirical measure of the normalized singular values of increasing rectangular submatrices of an infinite random matrix of independent elements. The limit is the limit as both dimensions grow large in some ratio. The matrix elements are required to have uniformly bounded central $2 + \delta$th moments, and the same means and variances within a row. The first section (relaxing the restriction on variances) proves any limit-in-distribution to be a constant measure rather than a random measure, establishes the existence of subsequences convergent in probability, and gives a criterion for almost-sure convergence. The second section proves the almost-sure limit to exist whenever the distribution of the row variances converges. It identifies the limit as a nonrandom probability measure which may be evaluated as a function of the limiting distribution of row variances and the dimension ratio. These asymptotic formulae underlie recently developed methods of probability plotting for principal components and have applications to multiple discriminant ratios and other linear multivariate statistics.

Limit theorems for sums of dependent random variables occurring in statistical mechanics

Abstract: We study the asymptotic behavior of partial sums S nfor certain triangular arrays of dependent, identically distributed random variables which arise naturally in statistical mechanics. A typical result is that under appropriate assumptions there exist a real number m, a positive real number λ, and a positive integer k so that (S n−nm)/n1−1/2k converges weakly to a random variable with density proportional to exp(−λ¦s¦ 2k/(2k)!). We explain the relation of these results to topics in Gaussian quadrature, to the theory of moment spaces, and to critical phenomena in physical systems.

Functional Limit Theorems for Dependent Variables

Abstract: The first result is a generalization of the classical results for independent random variables. The second result gives general conditions for convergence to processes which can be obtained from Brownian motion by a random change of time. This result is used to give a unified development of most of the martingale central limit theorems in the literature. An important aspect of our methods is that after the initial result is shown, we can avoid any further consideration of tightness.

Marcinkiewicz Laws and Convergence Rates in the Law of Large Numbers for Random Variables with Multidimensional Indices

Abstract: Consider a set of independent identically distributed random variables indexed by $Z^d_+$, the positive integer $d$-dimensional lattice points, $d \geqq 2$. The classical Kolmogorov-Marcinkiewicz strong law of large numbers is generalized to this case. Also, convergence rates in the law of large numbers are derived, i.e., the rate of convergence to zero of, for example, the tail probabilities of the sample sums is determined.

Essential Singularity in Percolation Problems and Asymptotic Behavior of Cluster Size Distribution

Abstract: It is rigorously proved that the analog of the free energy for the bond and site percolation problem on\(\mathbb{Z}^v \) in arbitrary dimensionΝ (Ν> 1) has a singularity at zero external field as soon as percolation appears, whereas it is analytic for small concentrations. For large concentrations at least, it remains, however, infinitely differentiable and Borel-summable. Results on the asymptotic behavior of the cluster size distribution and its moments, and on the average surface-to-size ratio, are also obtained. Analogous results hold for the cluster generating function of any equilibrium state of a lattice model, including, for example, the Ising model, but infinite-range andn-body interactions are also allowed.

On the limit theorems for random variables with values in the spaces L p (2≦p<∞)

Abstract: We prove that whenever B is an infinite dimensional Banach space, there exists a B-valued random variable X failing the Central Limit Theorem (in short the CLT) and such that IE∥X∥2=∞ but yet satisfying the Law of the Iterated Logarithm (in short the LIL) We obtain a new sufficient condition for the LIL in Hilbert space and we characterize the random variables with values in l p or L p with 2

Reduced $U$-Statistics and the Hodges-Lehmann Estimator

Abstract: A reduced $U$-statistic (of order 2) is defined as the sum of terms $f(X_i, X_j),$ where $f$ is a symmetric function, $(X_1, \cdots, X_N)$ are independent and identically distributed (i.i.d.) random variables (rv's), and $(i,j)$ are drawn from a restricted, though balanced, set of pairs. (A $U$-statistic corresponds to summation over all $(i, j)$ pairs.) A limit normal distribution is found for the reduced $U$-statistic, and it follows that estimates based on reduced $U$-statistics can have asymptotic efficiencies comparable with those based on $U$-statistics, even though the number of steps in computing a reduced $U$-statistic becomes asymptotically negligible in comparison with the number required for the corresponding $U$-statistic. As an illustration, a short-cut version of the Hodges-Lehmann estimator is defined, and its asymptotic properties derived, from a corresponding reduced $U$-statistic. A multivariate limit theorem is proved for a vector of reduced $U$-statistics, plus another result obtaining asymptotic normality even when $(i, j)$ are drawn from an unbalanced set of pairs. The present results are related to those of Blom.

Central limit theorems in D[0, 1]

Abstract: Let X be a stochastic process with sample paths in the usual Skorohod space D[0, 1]. For a sequence {Xn} of independent copies of X, let Sn=X1+⋯+Xn. Conditions which are either necessary or sufficient for the weak convergence of n−1/2(Sn−ESn) to a Gaussian process with sample paths in D[0, 1] are discussed. Stochastically continuous processe are considered separately from those with fixed discontinuities. A bridge between the two is made by a Decomposition central limit theorem.

Martingale central limit theorems and asymptotic estimation theory for multitype branching processes

Abstract: This paper considers the problem of estimating the growth rate ρ of a p-type Galton–Watson process {Z n }. To this end, a general approach of possible independent interest to central limit theorems for discrete-time branching processes is developed. The idea is to adapt martingale central limit theory to martingale difference triangular arrays indexed by the set of all individuals ever alive. Iterated logarithm laws are derived by similar methods. Asymptotic distribution results and the a.s. asymptotic behaviour are derived for a maximum likelihood estimator based upon all parent–offspring combinations in a given number N of generations, and for the estimator which depends on the total generation sizes only.

Limit theorems for weakly exchangeable arrays

Abstract: An array of random variables, indexed by a multidimensional parameter set, is said to be dissociated if the random variables are independent whenever their indexing sets are disjoint. The idea of dissociated random variables, which arises rather naturally in data analysis, was first studied by McGinley and Sibson(7). They proved a Strong Law of Large Numbers for dissociated random variables when their fourth moments are uniformly bounded. Silver man (8) extended the analysis of dissociated random variables by proving a Central Limit Theorem when the variables also satisfy certain symmetry relations. It is the aim of this paper to show that a Strong Law of Large Numbers (under more natural moment conditions), a Central Limit Theorem and in variance principle are consequences of the symmetry relations imposed by Silverman rather than the independence structure. To prove these results, reversed martingale techniques are employed and thus it is shown, in passing, how the well known Central Limit Theorem for U-statistics can be derived from the corresponding theorem for reversed martingales (as was conjectured by Loynes(6)).

Zone of influence models for competition in plantations

Abstract: BERAN, M. J. AND McCoy, J. J. (1974) Propagation through an anisotropic random medium. J. Math. Phys. 1S, 1901-1913. SUlTON, G. R. AND McCoy, J. J. (1976) Scattering of acoustic signals by inhomogeneities in a waveguide-a single scatter treatment. J. Acoust. Soc. Amer. 60, 833-839. SUlTON, G. R. AND McCoy, J. J. (1977) Spatial coherence of signals in randomly inhomogeneous waveguides-a multiple scatter theory. J. Math. Phys. 18, 1052-1057.

Limit Theorems for Multiply Indexed Mixing Random Variables, with Application to Gibbs Random Fields

Abstract: If $d$ is a fixed positive integer, let $\Lambda_N$ be a finite subset of $Z^d$, the lattice points of $\mathbb{R}^d$, with $\Lambda_N \uparrow Z^d$ and satisfying certain regularity properties. Let $(X_{N, Z})_{Z\in\Lambda_N}$ be a collection of random variables which satisfy a mixing condition and whose partial sums $X_N = \sum_{Z\in\Lambda_N} X_{N, Z}$ have uniformly bounded variances. Limit theorems, including a central limit theorem, are obtained for the sequence $X_N$. The results are applied to Gibbs random fields known to satisfy a sufficiently strong mixing condition.

Asymptotic properties of time domain gaussian estimators

Abstract: We consider a general non-linear multivariate time series model which can be parameterized by a finite and fixed number of parameters and which can be rewritten, if necessary, in a form such that the disturbances are stationary martingale differences. Given a series of discrete, equally spaced observations we prove the strong consistency and asymptotic normality of the Gaussian estimators of the parameters, the parameters possibly being subject to nonlinear constraints. Because the normal equations are usually highly non-linear it may be difficult to obtain explicit expressions for the Gaussian estimates. To overcome this problem we use a Gauss-Newton type algorithm to obtain a sequence of iterates which converge to, and have the same asymptotic properties as, the Gaussian estimates.

An Extended Martingale Invariance Principle

Abstract: In this note the conditions on an invariance principle for triangular arrays of random variables contained in an earlier paper are weakened. Random norming by functions which are not stopping times is permitted, the $L^2$-boundedness conditions on the maximum of the summands relaxed, and joint convergence with an arbitrary sequence of random elements of some other metric space proved.

Moment Inequalities for Sn Under General Dependence Restrictions, with Applications.

Abstract: : Consider a sum composed of a sequence of random variables and a sequence of constants. This paper establishes moment inequalities with dependence restrictions imposed upon the random variables but not depending upon the constants. A further inequality of more complicated form is also established. The dependence restrictions considered are either of the weak multiplicative type or of related types, namely exchangeable sequences and strongly mixing sequences. Three applications are developed. One treats the almost sure convergence of series under mild dependence restrictions and finite limit conditions. Secondly, an improved technique is presented for the problem of establishing the rate of convergence in the central limit theorem for simple linear rank statistics. Finally, the central limit theorem for strongly mixing summands is treated.

Some limit theorems for random fields

Abstract: We prove a central limit theorem with remainder and an iterated logarithm law for collections of mixing random variables indexed byZd,d≧1. These results are applicable to certain Gibbs random fields.

Continuity and the Central Limit Theorem for Random Trigonometric Series

Abstract: A Jensen type inequality on the non-decreasing rearrangement of a stochastic process enables an argument of 1-6] to be completed to yield a sufficient condition for the uniform convergence of random trigonometric series that is also necessary in many cases. In addition to extending the result in [6], in the case of random Fourier series, the sufficient condition is strictly weaker than the conditions of Salem and Zygmund [-11] and Kahane [9J. The inequality is also used to show that all pregaussian random Fourier series with independent terms satisfy the central limit theorem. Let f(x)>O, xe[0, 1]. For y > 0 define mz(y)=2{x: f ( x ) 0 let f ( x )=sup{y: ms(y)

The distribution of random determinants

Abstract: The exact distribution of fAnf , where An is a random determinant with independent and identically distributed exponential elements is given for the cases n = 2 and 3 . From the investigation of the behaviour of the density functions for these cases it is conjectured that for any fixed n, the probability density of [Anl for large values of the argument is the same as the density of (Y/n)n , where Y is a gamma random variable.