Showing papers on "Central limit theorem published in 1976"

Normal Approximation and Asymptotic Expansions

Abstract: Preface to the Classics Edition Preface 1. Weak convergence of probability measures and uniformity classes 2. Fourier transforms and expansions of characteristic functions 3. Bounds for errors of normal approximation 4. Asymptotic expansions-nonlattice distributions 5. Asymptotic expansions-lattice distributions 6. Two recent improvements 7. An application of Stein's method Appendix A.1. Random vectors and independence Appendix A.2. Functions of bounded variation and distribution functions Appendix A.3. Absolutely continuous, singular, and discrete probability measures Appendix A.4. The Euler-MacLaurin sum formula for functions of several variables References Index.

A Method for Simulating Stable Random Variables

Abstract: A new algorithm is presented for simulating stable random variables on a digital computer for arbitrary characteristic exponent α(0 < α ≤ 2) and skewness parameter β(-1 ≤ β ≤ 1). The algorithm involves a nonlinear transformation of two independent uniform random variables into one stable random variable. This stable random variable is a continuous function of each of the uniform random variables, and of α and a modified skewness parameter β' throughout their respective permissible ranges.

The Law of Large Numbers and the Central Limit Theorem in Banach Spaces

Abstract: Let $X_1, X_2, \cdots$ be independent random variables with values in a Banach space $E$. It is then shown that Chung's version of the strong law of large numbers holds, if and only if $E$ is of type $p$. If the $X_n$'s are identically distributed, then it is shown that the central limit theorem is valid, if and only if $E$ is of type 2. Similar results are obtained for vectorvalued martingales.

Vector linear time series models

Abstract: This paper presents proofs of the strong law of large numbers and the central limit theorem for estimators of the parameters in quite general finite-parameter linear models for vector time series. The estimators are derived from a Gaussian likelihood (although Gaussianity is not assumed) and certain spectral approximations to this. An important example of finite-parameter models for multiple time series is the class of autoregressive moving-average (ARMA) models and a general treatment is given for this case. This includes a discussion of the problems associated with identification in such models. LINEAR PROCESSES; VECTOR ARMA MODELS; IDENTIFICATION; LIMIT THEOREMS;

The Effect of Non-Normality on the Control Limits of X-Bar Charts

Abstract: The setting of limits on X-bar control charts, as well as other techniques employed in industrial statistics, are based on an assumption of normality justified by the central limit theorem. The theorem essentially states that, under general conditions, ..

The Asymptotic Distribution of Serial Covariances

Abstract: A central limit theorem is proved for the sample serial covariances of an ergodic, stationary, purely nondeterministic process whose linear innovations have their first four moments as for a sequence of independent random variables. The necessary and sufficient condition for the theorem is then that the spectra be square integrable.

Limit Theorems of Occupation Times for Markov Processes

Abstract: where u(f) is some normalizing function, has been investigated by many authors. A most general limit theorem was obtained by Darling and Kac [1], who also showed that, under suitable condition, the limit distribution must be Mittag-Leffler distribution. However, they confined themselves to the case where /(x) is nonnegative. C. Stone [5] derived a limit theorem for processes including 1 -dimensional diffusion processes with the infinitesimal generator -= -= — . It is not assumed that f ( x ) dm ax C is nonnegative, but it is essential that /(x) is not null-charged; \f(x)m(dx) *0. In this paper we study the case where f ( x ) is null-charged. If Xt is positively recurrent, the problem above can be reduced to the central limit theorem (see Tanaka [6]). A similar problem was treated by Dobrusin [2], who studied limit theorems for the 1 -dimensional simple random walk. The aim of this paper is to give a limit theorem for most general processes. Contrary to the case of [1], the limiting distribution is

Limit theorems for dissociated random variables

Abstract: Families of exchangeably dissociated random variables are defined and discussed. These include families of the form g(Y,, Y,, Y , Y,) for some function g of m arguments and some sequence Y, of i.i.d. random variables on any suitable space. A central limit theorem for exchangeably dissociated random variables is proved and some remarks on the closeness of the normal approximation are made. The weak convergence of the empirical distribution process to a Gaussian process is proved. Some applications to data analysis are discussed. CENTRAL LIMIT THEOREM; DISTANCE DISTRIBUTION; SIMILARITY MEASURE; TEST OF RANDOMNESS; TEST OF CLUSTERING; CLUSTER ANALYSIS; DEPENDENT RANDOM VARIABLES; WEAK CONVERGENCE; EMPIRICAL DISTRIBUTION FUNCTION; GAUSSIAN PROCESS; GRAPH COLOURING

Nonuniform Central Limit Bounds with Applications to Probabilities of Deviations

Abstract: For the distribution of the standardized sum of independent and identically distributed random variables, nonuniform central limit bounds are proved under an appropriate moment condition. From these theorems a condition on the sequence $t_n, n \in \mathbb{N}$, is derived which implies that $1 - F_n(t_n)$ is equivalent to the corresponding deviation of a normally distributed random variable. Furthermore, a necessary and sufficient condition is given for $1 - F_n(t_n) = o(n^{-c/2}t_n^{2 + c})$.

The Asymptotic Distribution of the Range and Other Functions of Partial Sums of Stationary Processes

Abstract: Let ℰn, n = 1, 2, ⋯ , be the net input in a reservoir during the nth period of time, and set S0 = 0, Sn = ℰ1 + … + ℰ n, = 1, 2, ⋯ . Many quantities of interest, such as range, first-passage times, and duration of deficit period, are functions of the partial sums Sn. In this paper it is pointed out that the functional central limit theorem, which has been previously used to obtain asymptotic results for independent and identically distributed ℰn, can be applied to a class of stationary sequences as well. To this class belong m-dependent, Markov, autoregressive, and autoregressive-moving average types of stationary processes.

Central limit theorem in a Banach space

Abstract: We describe the recent work by various authors on the central limit theorem in a Banach space E. Let {Xn} be a sequence of E-valued independent identically distributed random variables. X1 (or its distribution in E) is said to obey the CLT if the distributions of {n−1/2(X1+...+Xn)} converge weakly to a probability measure ν on E (which is necessarily Gaussian). Hoffmann-Jorgensen and Pisier have recently shown that every E-valued random variable X1 with e[X1]=0 and e[‖X1‖2]<∞ satisfies CLT ⇔ E is of “type 2”. Zinn has used this result to derive an earlier result of Jain and Marcus on CLT for C(S)-valued random variables. All these results and their proofs are included. Some recent work of Devary is also described where conditions are imposed on the distribution of X1 rather than on its modulus of continuity (as Jain and Marcus did in their main result).

Asymptotic inference for an Ising lattice

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.

A Central Limit Theorem for the Subadditive Process and Its Application to Products of Random Matrices

Abstract: The purpose of the present paper is to give a central limit theorem for subadditive process in the sense of J. F. C. Kingman (cf. [3], [4]). Throughout this article (O5 , P) denotes a probability space on which all random variables are defined. Let T be a measure preserving transformation in what follows. According to Kingman, a family (xsy9 s

Convergence en loi et lois du logarithme itéré pour les vecteurs gaussiens

Abstract: In this paper we prove a Strassen version of the law of the iterated logarithm for some sequences of weakly asymptotically independant Banach space valued gaussian random variables which converge in distribution, and we prove that the central limit theorem implies the functional form of the law of the iterated logarithm for the partial sums of certain Banach space valued gaussian sequences.

The central limit theorem for random fields

Abstract: A new variant of CLT is established for random elds de ned on Rd which are strictly stationary, with nite second moment and weakly dependent (comprising cases of positive or negative association). The summation domains grow in the van Hove sense. Simultaneously the indices of observations form more and more dense grids in these domains. Thus the e ect of combining two scaling procedures is studied. A statistical version of this CLT is also proved. Some stochastic models in Radiobiology based on dependent functional subunits are discussed as well.

A Two-Dimensional Functional Permutational Central Limit Theorem for Linear Rank Statistics

Abstract: Some two-dimensional time-parameter stochastic processes are constructed from a sequence of linear rank statistics by considering their projections on the spaces generated by the (double) sequence of anti-ranks. Under appropriate regularity conditions, it is shown that these processes weakly converge to Brownian sheets in the Skorokhod $J_1$-topology on the $D^2\lbrack 0, 1 \rbrack$ space. This unifies and extends earlier one-dimensional invariance principles for linear rank statistics to the two-dimensional case. The case of contiguous alternatives is treated briefly.

Asymptotic Results for Estimators in a Subcritical Branching Process with Immigration

Abstract: It is shown that certain estimators of the offspring and immigration means in a subcritical simple branching process with immigration are strongly consistent and obey the central limit theorem and law of the iterated logarithm under natural conditions.

On the Rate of Convergence in the Central Limit Theorem for Markov-Chains

Abstract: All results concerning the accuracy of the normal approximation for sums of not necessarily independent random variables assume Doeblin's condition or the condition of (p-mixing (see e.g. [1, 3, 5, 7, 9]). Both assumptions mean in some sense that the random variables are "asymptotically independent", and they are rarely fulfilled for Markov-chains. Using Doeblin's condition or the condition of (p-mixing the rate of convergence to the normal distribution obtained in some of the papers cited above is of order n -1/2. The authors do not know of any results on the accuracy of the normal approximation holding without such conditions. In this paper we prove under weak moment conditions that for Markov-chains the normal approximation is of order n -" for each c~ < 1/4.

Tendency towards normality of linear combinations of random variables

Abstract: IfX andY are two random variables with the same means and variances, thenX is said to be nearer normal thanY if the absolute values of its cumulants are smaller than the corresponding cumulants ofY. Using this definition, it is shown that a linear combination of a finite number of independent identically distributed random variables is always nearer normal than its constituents, but that this is not necessarily true if not-identically distributed or not-independent variables are used. Some consequences of the results are reached for the testing of normality of time series and for the assumptions frequently made by social scientists about the distribution of their data.

A Counterexample for Banach Space Valued Random Variables

Abstract: There exists a sequence of i.i.d. random variables taking values in the infinite dimensional Banach space $c_0$ satisfying the law of the iterated logarithm and failing to obey the central limit theorem.