Showing papers on "Central limit theorem published in 1983"
TL;DR: In this article, it was shown that the central limit theorem holds for some non-linear functionals of stationary Gaussian fields if the correlation function of the underlying field tends fast enough to zero.
412 citations
TL;DR: In this article, the authors extend these results considerably and prove central limit theorems and their rate of convergence (in the Prohorov metric and a Berry Esseen type theorem), functional central limit theorem and as approximation by a Brownian motion.
Abstract: Some probabilistic limit theorems for Hoeffding's U-statistic [13] and v Mises' functional are established when the underlying processes are not necessarily independent We consider absolutely regular processes [24] and processes (X
n)n≧1 which are uniformly mixing [14] as well as their time reversal (X
−n
)n≦−1, called uniformly mixing in both directions of time Many authors have weakened the hypothesis of independence in statistical limit theorems and considered m-dependent, Markov or weakly dependent processes; in particular for U statistics under weak dependence Sen [22] has considered *-mixing processes and derived a central limit theorem and a law of the iterated logarithm, while Yoshihara [26] proved central limit theorems and as results in the absolutely regular and uniformly mixing case Here we extend these results considerably and prove central limit theorems and their rate of convergence (in the Prohorov metric and a Berry Esseen type theorem), functional central limit theorems and as approximation by a Brownian motion Extensions to multisample versions and other extensions are briefly discussed
231 citations
TL;DR: In this paper, the limiting behavior of sums of functions of nearest neighbor distances is studied for an m-dimensional sample, and a central limit theorem and moment bounds for such sums, and an invariance principle for the empirical process of nearest neighbour distances are both established.
Abstract: : The limiting behavior of sums of functions of nearest neighbor distances is studied for an m dimensional sample. A central limit theorem and moment bounds for such sums, and an invariance principle for the empirical process of nearest neighbor distances are both established. As a consequence the asymptotic behavior of a practicable goodness of fit test is obtained based on nearest neighbor distances.
181 citations
TL;DR: In this paper, the field variables of the radiant surface for hadronic emission are distributed as gaussian random varibles (with k effective sources) as suggested by a typical central limit theorem argument.
108 citations
97 citations
TL;DR: For strong mixing sequences with error term o((t log log logt)1/2) as mentioned in this paper, a strong invariance principle was obtained by Berkes and Philipp (1979).
Abstract: We prove an estimate for the Prohorov-distance in the central limit theorem for strong mixing Banach space valued random variables. Using a recent variant of an approximation theorem of Berkes and Philipp (1979) we obtain as a corollary a strong invariance principle for absolutely regular sequences with error term
$$t^{\tfrac{1}{2} - \gamma }$$
. For strong mixing sequences we prove a strong invariance principle with error term o((t log logt)1/2).
86 citations
TL;DR: In this article, a small deviation theorem of a new form for the functional central limit theorem for partial sums of independent, identically distributed finite-dimensional random vectors was proved, and applied to obtain a functional form of the Chung-Jain-Pruitt law of the iterated logarithm which is also a strong speed of convergence theorem refining Strassen's invariance principle.
Abstract: We prove a small deviation theorem of a new form for the functional central limit theorem for partial sums of independent, identically distributed finite-dimensional random vectors. The result is applied to obtain a functional form of the Chung-Jain-Pruitt law of the iterated logarithm which is also a strong speed of convergence theorem refining Strassen's invariance principle.
83 citations
TL;DR: In this article, a class of empirical processes having the structure of $U$-statistics is considered, and the weak convergence of the processes to a continuous Gaussian process is proved in weighted sup-norm metrics stronger than the uniform topology.
Abstract: A class of empirical processes having the structure of $U$-statistics is considered. The weak convergence of the processes to a continuous Gaussian process is proved in weighted sup-norm metrics stronger than the uniform topology. As an application, a central limit theorem is derived for a very general class of non-parametric statistics.
74 citations
TL;DR: In this paper, a central limit theorem for the magnetization and energy densities of random variables satisfying the FKG inequalities is given, which is applicable to non-monotonic functions.
Abstract: A central limit theorem is given which is applicable to (not necessarily monotonic) functions of random variables satisfying the FKG inequalities. One consequence is convergence of the block spin scaling limit for the magnetization and energy densities (jointly) to the infinite temperature fixed point of independent Gaussian blocks for a large class of Ising ferromagnets whenever the susceptibility is finite. Another consequence is a central limit theorem for the density of thesurface of the infinite cluster in percolation models.
65 citations
TL;DR: In this paper, strong laws of large numbers concerning nonnegative random variables are obtained and then they are utilized to establish stability results, among other things, for sums of pairwise independent random variables and the range of random walks.
59 citations
TL;DR: In this paper, the central limit theorem is used to obtain a number of important results in random variable theory, including the density functions of some common probability distributions (lognormal, chi-square, and Student's t), the imposition of constraints on random variables, some sampling properties of the normal distribution, and the chi−square goodness-offit test.
Abstract: A theorem is presented which tells how to calculate the joint probability distribution of m random variables that have been defined as functional transformations of n given random variables (m, n≥1). Although the theorem involves Dirac delta functions and therefore has a rather formal appearance, it turns out to be surprisingly useful. It is used here to develop a number of important results in random variable theory, including: the central limit theorem, the density functions of some common probability distributions (lognormal, chi‐square, and Student’s‐t), the imposition of constraints on random variables, some sampling properties of the normal distribution, and the chi‐square goodness‐of‐fit test. Special attention is given to aspects of these topics that find fruitful application in physics. A broad conclusion seems to be that the theorem presented here provides a systematic way of obtaining many results in random variable theory which, although quite useful in physics, are normally found derived only in moderately advanced mathematics textbooks.
TL;DR: In this paper, the central limit theorem in B and the weak law of large numbers (for the sum of the squares of the random vectors) in another Banach lattice B(2) were shown to be equivalent.
Abstract: For B a type 2 Banach lattice, we obtain a relationship between the central limit theorem in B and the weak law of large numbers (for the sum of the squares of the random vectors) in another Banach lattice B(2). We then obtain some two-sided estimates for E∥Sn∥pwhich in lpspaces, 1≦p<∞, give n.a.s.c. for the weak law of large numbers. As a consequence of these estimates we also solve the domain of attraction problem in lp, p<2. Several examples and counterexamples are provided.
TL;DR: In this paper, under the conditions related to moment and dependence coefficients, it was shown that the central limit theorem for random fields satisfying some weak dependence condition has a rate of O(¦V¦−1/2(log σ)d) (m-dependent case).
Abstract: Let {X
a
;a∈Z
d} (d≧2) be a random field satisfying some weak dependence condition. For a finite subset V of Z
d
, set
$$S(V) = \sum\limits_{a \in V} {X_a } $$
. In this paper, under the conditions related to moment and dependence coefficients, we show that L
∞- and L
1-rates in the central limit theorem for S(V) are of order O(¦V¦−1/2(log¦V¦)d) (strong mixing case): O(¦V¦−1/2) (m-dependent case). Here ¦V¦ denotes the number of elements in V. The content of this paper is a negative answer to the conjecture of Prakasa Rao (Z. Wahrscheinlichkeitstheorie verw. Gebiete 58, 247–256 (1981)).
01 Jul 1983
TL;DR: Theorem for Physicists in the Theory of Random Variables (AD-A166 363) as discussed by the authors is an extension of the RVT theorem for the case of Dirac delta functions.
Abstract: : This paper consists of ten addenda to the article: A Theorem for Physicists in the Theory of Random Variables (AD-A166 363). Keywords: Dirac delta function, Gamma distribution, Normal distribution, Poisson distribution, Random variables, Random variable tranformation theorem, RVT theorem.
TL;DR: In this paper, the Chi-squared approximation of the distribution of a sum of independent random variables is studied and it is shown that the error of approximation is of order $n −1 −1/2 as $n \rightarrow \infty.
Abstract: We suggest several Chi squared approximations to the distribution of a sum of independent random variables, and derive asymptotic expansions which show that the error of approximation is of order $n^{-1}$ as $n \rightarrow \infty$. The error may be reduced to $n^{-3/2}$ by making a simple secondary approximation.
01 Jan 1983
TL;DR: In this paper, the authors present a set of properties of one-dimensional Markov random fields with a continuous state space and a test for multimodality based on kernel density estimates.
Abstract: 1. The asymptotic speed and shape of a particle system D. Aldous and J. Pitman 2. On doubly stochastic population processes M. S. Bartlett 3. On limit theorems for occupation times N. H. Bingham and J. Hawkes 4. The Martin boundary of two dimensional Ornstein-Uhlenbeck processes M. Cranston, S. Orey and U. Rosler 5. Green's and Dirichlet spaces for a symmetric Markov transition function E. B. Dynkin 6. On a theorem of Kabanov, Liptser and Sirjaev G. K. Eagleson and R. F. Gundy 7. Oxford Commemoration Ball J. M. Hammersley 8. Invariant measures and the q-matrix F. P. Kelly 9. The appearance of a multivariate exponential distribution in sojourn times for birth-death and diffusion processes J. T. Kent 10. Three unsolved problems in discrete Markov theory J. F. C. Kingman 11. The electrostatic capacity of an ellipsoid P. A. P. Moran 12. Stationary one-dimensional Markov random fields with a continuous state space F. Papangelou 13. A uniform central limit theorem for partial-sum processes indexed by sets R. Pyke 14. Multidimensional randomness B. D. Ripley 15. Some properties of a test for multimodality based on kernel density estimates B. W. Silverman 16. Criteria for rates of convergence of Markov chains, with application to queueing and storage theory R. L. Tweedie 17. Competition and bottle-necks P. Whittle.
TL;DR: In this paper, necessary and sufficient conditions for asymptotic normality of the sum of characteristics observed in a successive sample from a finite population were obtained for sampling fractions close to 0 or 1.
Abstract: The permutation distribution induced upon a finite population by the order of selection under successive sampling is closely related to the order statistics of independent exponentially distributed waiting times. This characterization is applied to obtain necessary and sufficient conditions for asymptotic normality of the sum of characteristics observed in a successive sample from a finite population. The necessary and sufficient conditions generalize previous results for simple random sampling without replacement, and apply to sampling fractions close to 0 or 1.
TL;DR: In this article, an expression for the distribution of a convolution of independent and identically distributed logistic random variables by directly inverting the characteristic function is obtained, which is closely approximated by a student'st distribution when both distribution are standardized.
Abstract: An expression is obtained for the distribution of a convolution of independent and identically distributed logistic random variables by directly inverting the characteristic function. This distribution is shown to be closely approximated by a student'st distribution when both distribution are standardized. Moreover, by showing that some of the analytic simplicity and statistical properties that are manifest in the single logistic also obtain in the convolution, an application of the convolution as a dose-response curve in the bio-assay problem is suggested.
TL;DR: In this article, the asymptotic behavior of the maximum of a set of independent, identically distributed Poisson random variables is investigated, and it is shown that the convergence from the maximum to the extreme two-point distribution is very slow and of an oscillatory nature.
Abstract: The asymptotic behaviour of the maximum of a set of independent, identically distributed Poisson random variables is investigated; the convergence of the maximum to the extreme two-point distribution is shown to be very slow and of an oscillatory nature.
TL;DR: Uniform rates of convergence in the Central Limit Theorem for associated random variables are given in this paper, where applications to the Ising Model and Diffusion and Gaussian processes are discussed.
Abstract: Uniform rates of convergence in the Central Limit Theorem for associated random variables are given. Applications to the Ising Model and Diffusion and Gaussian processes are discussed.
01 Jan 1983
TL;DR: In this article, central limit theorems are proved for some kernel-type estimators of probability density in the case where the observations form a strictly random sequence satisfying the ϱ-mixing condition with a certain logarithmic mixing rate.
TL;DR: In this paper, a histogram smoothed by a suitable family of lattice distributions is used as an estimator for an unknown probability density function, concentrated on a known interval.
Abstract: As an estimator\(\hat f_N \) for an unknown probability density functionf, concentrated on a known intervalI, one can use a histogram smoothed by a suitable family of lattice distributions. For such an estimator a uniform weak consistency result and a central limit theorem with an error bound are given. Further for the global deviation of\(\hat f_N \) fromf the asymptotic distribution is developed.
01 Sep 1983
TL;DR: In this paper, central limit theorems for U-statistics whose kernels depend on the size of the observed sample were developed for interpoint distance and large angle statistics. But these results were not applied to obtain results for the large angle statistic.
Abstract: Motivated by problems in the analysis of spatial data, central limit theorems are developed for U-statistics whose kernels depend on the size of the observed sample These theorems are then applied to obtain results for the interpoint distance statistic and the large angle statistic
TL;DR: In this article, the authors deal with approximation methods for the distribution of random sums, a subject being of high interest especially in actuarial mathematics (distribution of the total claim during a fixed time interval).
Abstract: This paper deals with approximation methods for the distribution of random sums, a subject being of high interest especially in actuarial mathematics (distribution of the total claim during a fixed time interval). Above all the authors intended to deliver rigid proofs for some propositions (such as Esscher and Edgeworth approximation) which are established in relevant articles frequently only in heuristic manner.
TL;DR: In this paper, the martingale proof of the Kolmogorov strong law of large numbers was shown to be weaker for the index set of random variables in Z^d+d+n.i.d.
Abstract: For simplicity, let $d = 2$ and consider the points $(n, m)$ in $Z^2_+$, with $\theta m \leq n \leq \theta^{-1}m$, where $0 < \theta < 1$. For i.i.d. random variables with this set as an index set we present a law of the iterated logarithm, strong laws of large numbers and related results. We also observe that (and try to explain why) the martingale proof of the Kolmogorov strong law of large numbers yields a weaker result for this index set than the classical proofs, whereas this is not the case if the index set is all of $Z^d_+, d \geq 1$.
TL;DR: In this article, the strictly stationary random sequences satisfying "absolute regularity" were constructed, which is a 0−1 instantaneous function of an aperiodic Markov chain with countable irreducible state space, such that n−2 var (X1 + ⋯ + Xn) approaches 0 arbitrarily slowly as n → ∞ and (X 1 + ε+ε+Xn) is partially attracted to every divisible law.
TL;DR: The central limit theorem for estimates of parameters which specify the covariance structure of a zero mean, stationary, Gaussian, discrete time series observed at unequally spaced times was proved in this paper.