On the asymptotic order of accuracy of Tikhonov regularization

In this paper, we obtain a necessary and sufficient condition for a system governed by Dirichlet and Neumann problems for a self-adjoint elliptic operator with an infinite number of variables to have an optimal control of the distributed type which is characterized by a set of inequalities.The authors are indebted to Professor J. L. Lions, President of IRIA/LABORIA, Paris, France, and Professor Jean Pierre Aubin, University of Paris IX, Dauphine, France, for their valuable comments, and also to the referees for their constructive suggestions.     

A note on the existence and asymptotic behavior of nonoscillatory solutions of fourth order quasilinear differential equations

This paper is concerned with nonoscillatory solutions of the fourth order quasilinear differential equation

The asymptotic equivalence of Bayes and maximum likelihood estimation

Let (X, $\text{A}$) be a measurable space, Θ ? $\text{R}$ an open interval and P_Ω ∥ $\text{A}$, Ω ? Θ, a family of probability measures fulfilling certain regularity conditions. Let $\text{Ω}{}_{\mathrm{n}}$ be the maximum likelihood estimate for the sample size n. Let λ be a prior distribution on Θ and let $\text{R}{}_{\mathrm{<mtext>n,</mtext><mtext>x</mtext>}}$ be the posterior distribution for the sample size n given $\text{x}\text{? X}{}^{\mathrm{n}}$. $\text{L:}\text{Θ}\text{× Θ →}\text{R}$ denotes a loss function fulfilling certain regularity conditions and T_n denotes the Bayes estimate relative to λ and L for the sample size n. It is proved that for every compact K ? Θ there exists c_K ≥ 0 such that

This theorem improves results of Bickel and Yahav [3], and Ibragimov and Has'minskii [4], as far as the speed of convergence is concerned.     

Integro-local theorems for sums of independent random vectors in the series scheme

Let S(n) = ξ(1)+?+ξ(n) be a sum of independent random vectors ξ(i) = ξ _(n)(i) with general distribution depending on a parameter n. We find sufficient conditions for the uniform version of the integro-local Stone theorem to hold for the asymptotics of the probability P(S(n) ∈ Δ[x), where Δ[x) is a cube with edge Δ and vertex at a point x.     

On the value distribution of the Matsumoto zeta-function

The Matsumoto zeta-function is defined by a polynomial Euler product and is a generalization of classical zeta-functions. In the paper a discrete limit theorem in the sense of the weak convergence of probability measures on the complex plane for the Matsumoto zeta-function is proved.     

Local asymptotic normality and asymptotical minimax efficiency of the MLE under random censorship

Here we study the problems of local asymptotic normality of the parametric family of distributions and asymptotic minimax efficient estimators when the observations are subject to right censoring. Local asymptotic normality will be established under some mild regularity conditions. A lower bound for local asymptotic minimax risk is given with respect to a bowl-shaped loss function, and furthermore a necessary and sufficient condition is given in order to achieve this lower bound. Finally, we show that this lower bound can be attained by the maximum likelihood estimator in the censored case and hence it is local asymptotic minimax efficient.     

Hájek-Šidák approach to the asymptotic distribution of multivariate rank order statistics

Let X_α = (X_1α,…, X_pα), 1 ≤ α ≤ N_ν, ν ≥ 1 be N_ν independent observations from a density function f(x) where x ∈ R^p, the p-dimensional real space. Let R_νjα denote the rank of X_jα in the ordered array of X_j1 ,…, X_jNν; 1 ≤ j ≤ p and consider the multivariate rank order statistics

$\text{T}{}_{\mathrm{vj}}\text{=}\text{∑}\text{α = 1}\text{N}{}_{\mathrm{v}}\text{C}{}_{\mathrm{y\alpha }}\text{a}{}_{\mathrm{vj}}\text{(R}{}_{\mathrm{vj\alpha }}\text{),}$

where the constants, c_να, 1 ≤ α ≤ N_ν satisfy the Noether condition and the scores, a_νj(α), 1 ≤ j ≤ p, 1 ≤ α ≤ N_ν converge as ν → ∞, for each j, in quadratic mean to a nonconstant, square integrable function π_j(u), 0 < u < 1. Under the hypothesis of randomness, the joint asymptotic conditional and uncoditional normality of the statistics T_νj, 1 ≤ j ≤ p is established. Further, under mild conditions on the underlying density functions and assuming contiguous location shift alternatives, the joint asymptotic normality of these statistics is also established.     

任意随机变量序列泛函及其随机变换的一类极限定理

采用鞅方法研究对任意随机变量序列普遍成立的强极限定理.并作为推论得到了m阶马氏过程,鞅序列,鞅差序列,独立随机变量序列的一类强极限定理.并把赌博系统的随机变换概念推广到任意随机变量序列的情况,得到任意随机变量序列随机选择与公平比的若干极限定理.     

Value distribution of the Lerch zeta-function with algebraic irrational parameter. II

In this paper, we obtain a joint limit theorem in the sense of weak convergence of probability measures on the complex plane for Lerch zeta-functions with algebraic irrational parameters and give an explicit form of the limit measure. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 482–496, October–December, 2007.     

Tests of elliptical symmetry and the asymptotic tail behavior of the statistics

In this paper, some test statistics Of Kolmogorov type and Cramervon Mises type based on projection pursuit technique are proposed for testing the sphericity problem of a high-dimensional distribution. The limiting distributions of the test statistics are derived under the null hypothesis. The asymptotic properties of Bootstrap approximation are investigated and the tail behaviors of the statistics are studied.     

Chaos expansion and asymptotic behavior of the Pareto distribution

We give the chaos expansion of a random variable with Pareto distribution and we analyze, by using the Malliavin calculus, the convergence in the distribution of a sequence of random variable with Pareto distribution toward the standard exponential law.     

Boundedness, asymptotic behavior and oscillatory properties for the models of single-species population growth with impulsive effect

1. IntroductionThe models of single-species population growth can be described as the following ordinary differelltial equations(IVP) X' ~ f(x,K(t)), x(to) = xo 2 0, (1.1)where K(t) > 0 is the carrying capacity of the environment. Much has been done underthe assumption that K(t) is either constant or changed periodicallyll-3]. However, onemay easily visualize in nature where abruput changes such as harvesting, and disajsters mayoccur. Consequently, we consider such model with impulses. It …     

A central limit theorem and higher order results for the angular bispectrum

The angular bispectrum of spherical random fields has recently gained an enormous importance, especially in connection with statistical inference on cosmological data. In this paper, we analyze its moments and cumulants of arbitrary order and we use these results to establish a multivariate central limit theorem and higher order approximations. The results rely upon combinatorial methods from graph theory and a detailed investigation for the asymptotic behavior of coefficients arising in matrix representation theory for the group of rotations SO(3). I am very grateful to an associate editor and two referees for many useful comments, and to M. W. Baldoni and P. Baldi for discussions on an earlier version.     

Local Limit Theorems for Sums of Power Series Distributed Random Variables and for the Number of Components in Labelled Relational Structures

A Local limit theorem for the distribution of the number of components in random labelled relational structures of size n (e.g., a type of random graphs on n vertices, random permutations of n elements, etc.) is proved as n→∞. The case when the corresponding exponential generating functions diverge at their radii of convergence is considered.     

整值随机变量序列发生次数与条件概率和的等价性

在没有独立性、平稳性和相依性假设的条件下,利用分析方法讨论了整值随机变量序列取某个数值的次数与随机变量序列取该数值的条件概率和的等价性问题,建立并证明了若干强极限定理.     

On tree census and the giant component in sparse random graphs

For each of the two models of a sparse random graph on n vertices, G(n, # of edges = cn/2) and G(n, Prob (edge) = c/n) define t_n(k) as the total number of tree components of size k (1 ≤ k ≤ n). the random sequence {[t_n(k) - nh(k)]n^?1/2} is shown to be Gaussian in the limit n →∞, with h(k) = k^k?2c^k?1e^?kc/k! and covariance function being dependent upon the model. This general result implies, in particular, that, for c> 1, the size of the giant component is asymptotically Gaussian, with mean nθ(c) and variance n(1 ? T)^?2(1 ? 2Tθ)θ(1 ? θ) for the first model and n(1 ? T)^?2θ(1 ? θ) for the second model. Here Te^?T = ce^?c, T<1, and θ = 1 ? T/c. A close technique allows us to prove that, for c < 1, the independence number of G(n, p = c/n) is asymptotically Gaussian with mean nc^?1(β + β²/2) and variance n[c^?1(β + β²/2) ?c^?2(c + 1)β²], where βe^β = c. It is also proven that almost surely the giant component consists of a giant two-connected core of size about n(1 ? T)β and a “mantle” of trees, and possibly few small unicyclic graphs, each sprouting from its own vertex of the core.     

求无穷级数和以及多重积分极限的概率方法

文 [1 ]举例说明了概率思想在求无穷级数的和以及多重积分极限方面的应用 ,本文将对文 [1 ]中的例 1加以推广 ,拓展这一类题的解题思路 ,并对例 2用不同的概率方法加以证明 ,从而使数学分析与概率统计的有关知识联系起来 ,达到知识的融汇贯通 .     

Effect of the initial estimator on the asymptotic behavior of one-step M-estimator

For a general (real) parameter, let M _nbe the M-estimator and M _n ⁽¹⁾ be its one-step version (based on a suitable initial estimator M _n ⁽⁰⁾). It is known that, under certain regularity conditions, n(M _n ⁽¹⁾-M _n)=O _p(1). The asymptotic distribution of n(M _n ⁽¹⁾-M _n) is studied; it is typically non-normal and it reveals the role of the initial estimator M _n ⁽⁰⁾.Work of this author was partially supported by the Office of Naval Research, Contract No. N00014-83-K-0387     

Estimating the asymptotic dispersion of theL 1 median

A simple estimate of the asymptotic dispersion matrix of theL ₁ median is proposed and its rate of convergence is studied.Research supported in part by a grant of the Indian Statistical Institute.