On the Absolute Continuity of Gaussian Measures on Locally Compact Groups

This note discusses the topological necessary and sufficient conditions for a locally compact connected group to admit a Gaussian measure that is absolutely continuous with respect to Haar measure.     

谱任意的符号模式矩阵

一个n阶符号模式矩阵A称为是谱任意的,如果对任意的实系数n次首1多项式r(x),在A的定性矩阵类Q(A)中至少存在一个实矩阵B,使得B的特征多项式是r(x),文中证明了当n为奇数时n阶谱任意符号模式矩阵是存在的。     

紧拓扑半群上概率测度卷积序列的极限性质

本文讨论紧拓扑半群上概率测度卷积序列的若干重要极限性质．在第１节中,我们讨论测度集的代数结构与其支撑集代数结构的关系．第２节的定理１,通过支撑集的代数结构给出组合收敛测度序列的一个极限定理．在定理２中我们讨论独立同分布时的情形,建立了一类紧半群上的Ｋａｗａｄａ－Ｉｔ型结果．这些定理推广了紧群、紧交换半群、紧Ｌ－Ｘ半群上一些相应的结论．     

New matrix partial order based on spectrally orthogonal matrix decomposition

We investigate partial orders on the set of complex square matrices and introduce a new order relation based on spectrally orthogonal matrix decompositions. We also establish the relation of this concept with the known orders.     

有缺失数据的条件独立正态母体中参数的最优同变估计

在缺失数据机制是可忽略的假设下,导出了有单调缺失数据的条件独立正态模型中协方差阵和精度阵的Cholesky分解的最大似然估计和无偏估计.通过引入一类特殊的变换群并在更广义的损失下,获得了其最优同变估计.这表明最大似然估计和无偏估计是非容许的.最后,通过数值模拟验证了相关结果的有效性.     

Maxima of entries of Haar distributed matrices

Let _n=(_ij) be an n×n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also W_n:=max_1i,jn|_ij|. We obtain the limiting distribution and a strong limit theorem on W_n. A tool has been developed to prove these results. It says that up to n/( log n)² columns of _n can be approximated simultaneously by those of some Y_n=(y_ij) in which y_ij are independent standard normals. Similar results are derived also for the unitary group U(n), the special orthogonal group SO(n), and the special unitary group SU(n).Mathematics Subject Classification (2000):15A52, 60B10, 60B15, 60F10     

Itô-Wiener Chaos Expansion with Exact Residual and Correlation, Variance Inequalities

We give a formula of expanding the solution of a stochastic differential equation (abbreviated as SDE) into a finite Itô-Wiener chaos with explicit residual. And then we apply this formula to obtain several inequalities for diffusions such as FKG type inequality, variance inequality and a correlation inequality for Gaussian measure. A simple proof for Houdré-Kagan's variance inequality for Gaussian measure is also given.     

Analog of heat equation for gaussian measure of a ball in Hilberrt space

If _a,T is a Gaussian measure on a Hilbert space with meana and covariance operatorT, andr is a fixed positive number, then the functiong(a,T)= _a,T {xr} satisfies the equation

Measures of Non-compactness of Operators on Banach Lattices

[Indag. Math.(N.S.) 2(2) (1991), 149–158; Uspehi Mat. Nauk 27(1(163)) (1972), 81–146] used representation spaces to study measures of non-compactness and spectral radii of operators on Banach lattices. In this paper, we develop representation spaces based on the nonstandard hull construction (which is equivalent to the ultrapower construction). As a particular application, we present a simple proof and some extensions of the main result of [J. Funct. Anal. 78(1) (1988), 31–55] on the monotonicity of the measure of non-compactness and the spectral radius of AM-compact operators. We also use the representation spaces to characterize d-convergence and discuss the relationship between d-convergence and the measures of non-compactness.     

L2(C,μ)中的一组Haar基及相关收敛性

本文研究经典三分Cantor集C上的平方可积空间L2(C,μ).利用投影算子的相关结论,证明了此空间上存在一组Haar型规范正交基,进而分析了L2(C,μ)中任意元素关于此基的Fourier/Haar展开,并讨论了任意元素关于此级数展开的相关收敛性.     

Bayes estimation of number of signals

Bayes estimation of the number of signals, q, based on a binomial prior distribution is studied. It is found that the Bayes estimate depends on the eigenvalues of the sample covariance matrix S for white-noise case and the eigenvalues of the matrix S ₂ (S ₁+A)^–1 for the colored-noise case, where S ₁ is the sample covariance matrix of observations consisting only noise, S ₂ the sample covariance matrix of observations consisting both noise and signals and A is some positive definite matrix. Posterior distributions for both the cases are derived by expanding zonal polynomial in terms of monomial symmetric functions and using some of the important formulae of James (1964, Ann. Math. Statist., 35, 475–501).     

On the effectivization of the universality theorem for the Lerch zeta-function

In the paper one way of the effectivization of the universality theorem for the Lerch zeta-function is discussed. It is based on the estimate of the rate of convergence in a limit theorem in the space of analytic functions. Partially supported by the Lithuanian State Science and Studies Foundation. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 172–178, April–June, 2000. Translated by A. Laurinčikas     

Operator-decomposability of Gaussian measures on separable Banach spaces

An operator-decomposable Gaussian measure on a separable Banach space can be factorized into a convolution product of a strongly operator-decomposable Gaussian measure and an operator-invariant Gaussian measure (with respect to the same operator). An example for this very factorization is discussed in some detail. In particular it is shown that a strongly operator-decomposable Gaussian measure need not necessarily be supported by the contraction subspace of the operator involved. Finally, the decomposability semigroup of a Gaussian measure turns out to be convex; and the corresponding invariance semigroup belongs to its extreme boundary.     

Weak Mixing of Random Walks on Groups

Let G be a locally compact -compact group with right Haar measure m and a regular probability measure on G. We say that is weakly mixing if for all gL (G) and all fL ¹(G) with fdm=0 we have n ^{–1 n} _k=1| ^k *f,g|0. We show that is weakly mixing if and only if is ergodic and strictly aperiodic. To prove this we use and prove some results about unimodular eigenvalues for general Markov operators.     

加法与乘法下的模糊熵

由于模糊集关于普通加法与乘法运算构成的代数系统是分配格、布尔代数,这就使得模糊集的应用受到一定的限制,为了扩展模糊集的应用,本文引进了模糊集的加法与乘法运算,接着研究了模糊集关于加法与乘法的运算性质以及模糊熵,距离测度,相似性测度在集合关于加法运算,乘法运算下的一些性质,并且对加法与乘法下的模糊熵与普通模糊熵作了对比.