Asymptotic Expansions in the Compound Poisson Limit Theorem

A triangular array of independent infinitesimal integer-valued random variables is considered. Asymptotic expansions for the probability distributions of sums of these variables are investigated in the case of the limiting compound Poisson laws.     

Asymptotic Expansions for the Distribution of the Maximum of Gaussian Random Fields

Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ^N . They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian field above an high level inside S and on the border S. Depending on the geometry of the border we give up to N+1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown.     

Sharp Error Bounds for Asymptotic Expansions of the Distribution Functions for Scale Mixtures

Let Z be a random variable with the distribution function G(x) and let s be a positive random variable independent of Z. The distribution function F(x) of the scale mixture X = sZ is expanded around G(x) and the difference between F(x) and its expansion is evaluated in terms of a quantity depending only on G and the moments of the powers of the variable of the form s/gr - 1, where (> 0) and (= ±1) are parameters indicating the types of expansion. For = -1, the bound is sharp under some extra conditions. Sharp bounds for errors of the approximations of the scale mixture of the standard normal and some gamma distributions are given either by analysis ( = -1) or by numerical computation ( = 1).     

Asymptotic Distribution of a Kind of Dirichlet Distribution

The Dirichlet distribution that we are concerned with in this paper is very special, in which all parameters are different from each other. We prove that the asymptotic distribution of this kind of Dirichlet distributions is a normal distribution by using the central limit theorem and Slutsky theorem.     

渐近正态随机变量函数的极限分布

基于渐近正态随机变量,导出随机变量函数极限分布的两个一般性理论结果.作为应用,证明了渐近正态随机变量一系列具体函数的极限分布,其中包括泊松随机变量平方根的渐近正态性,以及随机变量部分和在正则化常数是随机变量情况下的渐近正态性.     

Asymptotic Expansions of Large Deviations for Sums of Nonidentically Distributed Random Variables

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in Cramer zones and Linnik power zones for the distribution function of sums of independent nonidentically distributed random variables (r.v.). In this scheme of summation of r.v., the results are obtained first by mainly using the general lemma on large deviations considering asymptotic expansions for an arbitrary r.v. with regular behaviour of its cumulants [11]. Asymptotic expansions in the Cramer zone for the distribution function of sums of identically distributed r.v. were investigated in the works [1,2]. Note that asymptotic expansions for large deviations were first obtained in the probability theory by J. Kubilius [3].     

Asymptotic Improvement of the Usual Confidence Set in a Multivariate Normal Distribution with Unknown Variance

We consider confidence sets for the mean of a multivariate normal distribution with unknown covariance matrix of the formσ²I. The coverage probability of the usual confidence set is shown to be improved asymptotically by centering at a shrinkage estimator.     

Complete Asymptotic Expansions for the Chlodovsky Polynomials

The purpose of this article is to study the local rate of convergence of the Chlodovsky operators (C_nf)(x). As the main results, we investigate their asymptotic behaviour and derive the complete asymptotic expansions of these operators. All the coefficients of n^?k (k = 1, 2,…) are calculated in terms of the Stirling numbers of first and second kind. We mention that analogous results for the Bernstein polynomials can be found in Lorentz [2 G. G. Lorentz ( 1953 ). Bernstein Polynomials . University of Toronto Press , Toronto . [Google Scholar]].     

广义高斯分布的参数估计及其收敛性质

广义高斯分布是一类以Gaussian分布、Laplacian分布为特例的对称分布 ,它在信号处理和图像处理等领域都有广泛的应用 .本文采用矩估计方法讨论广义高斯分布的形状参数和尺度参数的估计问题 ,首先导出了矩和参数的关系表达式 ,然后由此提出参数估计方法 ,并对参数估计的收敛性质进行了分析 ,最后利用模拟实验对本文所提方法进行了验证 .     

Krawtchouk多项式零点的渐近展开及其误差限

Krawtchouk多项式在现代物理学中有着广泛应用．基于Li和Wong的结果,利用Airy函数改进了Krawtchouk多项式的渐近展开式,而且得到了一个一致有效的渐近展开式A·D2进一步,利用Airy函数零点的性质,推导出了Krawtchouk多项式零点的渐近展开式,并讨论了其相应的误差限．同时还给出了Krawtchouk多项式和其零点的渐近性态,它优于Li和Wong的结果．     

Asymptotic expansions of the distributions of some test statistics for Gaussian ARMA processes

Let {X_t} be a Gaussian ARMA process with spectral density f_θ(λ), where θ is an unknown parameter. The problem considered is that of testing a simple hypothesis H:θ = θ₀ against the alternative A:θ ≠ θ₀. For this problem we propose a class of tests , which contains the likelihood ratio (LR), Wald (W), modified Wald (MW) and Rao (R) tests as special cases. Then we derive the χ² type asymptotic expansion of the distribution of T up to order n⁻¹, where n is the sample size. Also we derive the χ² type asymptotic expansion of the distribution of T under the sequence of alternatives A_n: θ = θ₀ + /√n, ε > 0. Then we compare the local powers of the LR, W, MW, and R tests on the basis of their asymptotic expansions.     

Asymptotic Expansions of Transition Densities for Hybrid Jump-diffusions

A class of hybrid jump diffusions modulated by a Markov chain is considered in this work.Themotivation stems from insurance risk models,and emerging applications in production planning and wirelesscommunications.The models are hybrid in that they involve both continuous dynamics and discrete events.Under suitable conditions,asymptotic expansions of the transition densities for the underlying processes aredeveloped.The formal expansions are validated and the error bounds obtained.     

多维局部平稳高斯过程最大值的联合渐近分布

$\{(X_1(t),\cdots,X_p(t)),0\leq t\leq T\}$为$p$维局部平稳高斯过程, 具有渐近中心化的均值$m_k(t)$和常数的方差, $M_k(T)=\sup\{X_k(t),0\leq t\leq T\},\;k=1,\cdots,p$, 当$T\rightarrow\infty$时, 本文在一定条件下获得了$M(T)=(M_1(T),\cdots,M_p(T))$的联合渐近分布.     

Uniform Convergence of Wavelet Expansions of Gaussian Random Processes

New results on uniform convergence in probability for the most general classes of wavelet expansions of stationary Gaussian random processes are given.     

Asymptotic Expansions for Large Deviation Probabilities of Noncentral Generalized Chi-Square Distributions

Asymptotic expansions for large deviation probabilities are used to approximate the cumulative distribution functions of noncentral generalized chi-square distributions, preferably in the far tails. The basic idea of how to deal with the tail probabilities consists in first rewriting these probabilities as large parameter values of the Laplace transform of a suitably defined function f_k; second making a series expansion of this function, and third applying a certain modification of Watson's lemma. The function f_k is deduced by applying a geometric representation formula for spherical measures to the multivariate domain of large deviations under consideration. At the so-called dominating point, the largest main curvature of the boundary of this domain tends to one as the large deviation parameter approaches infinity. Therefore, the dominating point degenerates asymptotically. For this reason the recent multivariate asymptotic expansion for large deviations in Breitung and Richter (1996, J. Multivariate Anal.58, 1–20) does not apply. Assuming a suitably parametrized expansion for the inverse g⁻¹ of the negative logarithm of the density-generating function, we derive a series expansion for the function f_k. Note that low-order coefficients from the expansion of g⁻¹ influence practically all coefficients in the expansion of the tail probabilities. As an application, classification probabilities when using the quadratic discriminant function are discussed.     

普阿松分布的极限分布

本文主要阐明概率论中三个重要分布之间的关系 ,重点证明了普阿松分布的极限分布是正态分布     

负相伴样本情形线性指数分布参数的经验Bayes检验问题

讨论了负相伴样本情形线性指数分布参数的经验Bayes(EB)单侧检验问题.利用概率密度函数的核估计构造了参数的经验Bayes单侧检验函数,在适当的条件下证明了所提出的经验Bayes检验函数的渐近最优(a.o.)性并获得了其收敛速度.最后给出一个有关主要结果的例子.     

Asymptotic Expansion of the Distribution Density Function for the Sum of Random Variables in the Series Scheme in Large Deviation Zones

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in the Cramer zone and Linnik power zones for the distribution density function of sums of independent random variables in a triangular array scheme. The result was obtained using general Lemma 6.1 of Saulis and Statuleviius in Limit Theorems for Large Deviations (Kluwer, 1991) and joining the methods of characteristic functions and cumulants. The work extends the theory of sums of random variables and in a special case, improves S. A.Book's results on sums of random variables with weights.     

The Logarithmic Asymptotic Expansions for the Norms of Evaluation Functionals

Let μ be a compactly supported finite Borel measure in ℂ, and let Π_n be the space of holomorphic polynomials of degree at most n furnished with the norm of L ²(μ). We study the logarithmic asymptotic expansions of the norms of the evaluation functionals that relate to polynomials p ∈ Π_n their values at a point z ∈ ℂ. The main results demonstrate how the asymptotic behavior depends on regularity of the complement of the support of μ and the Stahl-Totik regularity of the measure. In particular, we study the cases of pointwise and μ-a.e. convergence as n → ∞.Original Russian Text Copyright © 2005 Dovgoshei A. A., Abdullaev F., and Kucukaslan M.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 774–785, July–August, 2005.