Central limit theorem for quadratic forms for sparse tables

Sufficient conditions for asymptotic normality for quadratic forms in {n_t − np_t} are given, where {n_t} are the observed counts with expected cell means {np_t}. The main result is used to derive asymptotic distributions of many statistics including the Pearson's chi-square.     

McKean-Vlasov limit for interacting random processes in random media

We apply large-deviation theory to particle systems with a random mean-field interaction in the McKean-Vlasov limit. In particular, we describe large deviations and normal fluctuations around the McKean-Vlasov equation. Due to the randomness in the interaction, the McKean-Vlasov equation is a collection of coupled PDEs indexed by the state space of the single components in the medium. As a result, the study of its solution and of the finite-size fluctuation around this solution requires some new ingredient as compared to existing techniques for nonrandom interaction.     

On the structure of random plane-oriented recursive trees and their branches

This paper is an investigation of the structural properties of random plane-oriented recursive trees and their branches. We begin by an enumeration of these trees and some general properties related to the outdegrees of nodes. Using generalized Pólya urn models we study the exact and limiting distributions of the size and the number of leaves in the branches of the tree. The exact distribution for the leaves in the branches is given by formulas involving second-order Eulerian numbers. A martingale central limit theorem for a linear combination of the number of leaves and the number of internal nodes is derived. The distribution of that linear combination is a mixture of normals with a beta distribution as its mixing density. The martingale central limit theorem allows easy determination of the limit laws governing the leaves in the branches. Furthermore, the asymptotic joint distribution of the number of nodes of outdegree 0, 1 and 2 is shown to be trivariate normal. © 1993 John Wiley & Sons, Inc.     

Multivariate Variational Inequalities and the Central Limit Theorem

Multivariate variational inequalities are obtained in terms of thew-functions and the trace of a Fisher-type information matrix. In consequence of these inequalities, the multivariate central limit theorem arises in the sense of the total variation.     

Index Reduction Formulas for Twisted Flag Varieties, II

This is a continuation of the determination begun in K-Theory 10 (1996), 517–596, of explicit index reduction formulas for function fields of twisted flag varieties of adjoint semisimple algebraic groups. We give index reduction formulas for the varieties associated to the classical simple groups of outer type A _n-1 and D _n, and the exceptional simple groups of type E ₆ and E ₇. We also give formulas for the varieties associated to transfers and direct products of algebraic groups. This allows one to compute recursively the index reduction formulas for the twisted flag varieties of any semi-simple algebraic group.     

Hölderian Invariance Principle for Triangular Arrays of Random Variables

In this paper, we extend the Hölderian invariance principle of Lamperti [6] to the case of partial-sum processes based on a triangular array of row-wise independent random variables. As an application, we obtain necessary and sufficient conditions for the almost sure (resp. in probability) weak Hölder convergence of partial-sum processes based on bootstrapped samples.     

正四面体套中心构型的存在唯一性

Two cases of the nested configurations in R^3 consisting of two regular quadrilaterals are discussed. One case of them do not form central configuration, the other case can be central configuration. In the second case the existence and uniqueness of the central configuration are studied. If the configuration is a central configuration, then all masses of outside layer are equivalent, similar to the masses of inside layer. At the same time the following relation between r(the ratio of the sizes) and mass ratio b = m/m must be satisfied b=24(3的立方根)（3r^2 2r 3）^-3/2-8（1-r）|1-r|^-3-3(6r的立方根)/24(3的立方根)（3 r）（3r^2 2r 3）^-3/2-8r（1-r）|1-r|^-3-3(6r^-2的立方根)in which the masses at outside layer are not less than the masses at inside layer, and the solution of this kind of central configuration is unique for the given ratio (b) of masses.     

Computer-Generated Holograms Based on the Three-Dimensional Fourier Spectrum

A method to synthesize a computer-generated hologram (CGH) of real-existing objects from projection images is proposed. Different from other similar methods, our method enables the synthesis of a CGH with only one-dimensional (1-D) mechanical scanning of the objects. Our method is connected with the three-dimensional (3-D) Fourier spectrum of the objects by the 3-D central slice theorem (CST). Two efficient recording techniques for projection images and numerical experiments to verify our principle is discussed. A comparison between the two techniques is also presented from the viewpoint of diffraction efficiency. © 2005 The Optical Society of Japan     

Mann-Whitney test for associated sequences

Let {X ₁, ...,X _m } and {Y ₁, ...,Y _n } be two samples independent of each other, but the random variables within each sample are stationary associated with one dimensional marginal distribution functionsF andG, respectively. We study the properties of the classical Wilcoxon-Mann-Whitney statistic for testing for stochastic dominance in the above set up.     

Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn2, let (μ^x_τ, n)_τ0be the distributions of the Brownian motion on the unit sphereS^{n n+1}starting in some pointxSⁿ. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμ^x_τ, nto the uniform distributionU_nonSⁿforτ→∞ depending on the dimensionn. We show that,[formula]forτ_n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμ^x_τ, nby measures which are known explicitly via Poisson kernels onSⁿ, and which tend, after suitable projections and dilatations, to normal distributions on forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.