On some results of M. I. Gordin: A clarification of a misunderstanding

M. I. Gordin proved a central limit theorem for some strictly stationary strongly mixing random sequences without the assumption of finite second moments. Because of a series of misunderstandings, his own correct formulation of the theorem has been essentially ignored, while an incorrect formulation has been discussed extensively and attributed to him in many references. This note explains in detail what has happened, in the hope of clearing up the misunderstandings.     

A Central Limit Theorem for Non-stationary Strongly Mixing Random Fields

In this paper we extend a central limit theorem of Peligrad for uniformly strong mixing random fields satisfying the Lindeberg condition in the absence of stationarity property. More precisely, we study the asymptotic normality of the partial sums of uniformly \(\alpha \)-mixing non-stationary random fields satisfying the Lindeberg condition, in the presence of an extra dependence assumption involving maximal correlations.     

The central limit theorem for Markov chains started at a point

 The aim of this paper is to prove a central limit theorem and an invariance principle for an additive functional of an ergodic Markov chain on a general state space, with respect to the law of the chain started at a point. No irreducibility assumption nor mixing conditions are imposed; the only assumption bears on the growth of the L ² -norms of the ergodic sums for the function generating the additive functional, which must be with . The result holds almost surely with respect to the invariant probability of the chain. Received: 17 October 2001 / Revised version: 5 April 2002 / Published Online: 24 October 2002 Mathematics Subject Classification (2000): 60F05, 60J05     

Central limit theorems under weak dependence

This article is motivated by a central limit theorem of Ibragimov for strictly stationary random sequences satisfying a mixing condition based on maximal correlations. Here we show that the mixing condition can be weakened slightly, and construct a class of stationary random sequences covered by the new version of the theorem but not Ibragimov's original version. Ibragimov's theorem is also extended to triangular arrays of random variables, and this is applied to some kernel-type estimates of probability density.     

A splitting theorem for infinite dimensional Alexandrov spaces with nonnegative curvature and its applications

We prove a splitting theorem for Alexandrov space of nonnegative curvature without properness assumption. As a corollary, we obtain a maximal radius theorem for Alexandrov spaces of curvature bounded from below by 1 without properness assumption. Also, we provide new examples of infinite dimensional Alexandrov spaces of nonnegative curvature.     

Uniqueness Theorems for Generalized Haar Systems

A uniqueness theorem and a recovery theorem for the coefficients of series in generalized Haar systems are proved under the assumption that the series converge in measure and satisfy a certain necessary condition on the distribution function of the majorant of partial sums.     

A functional central limit theorem for strongly mixing sequences of random variables

Summary We prove a functional central limit theorem for a class of strongly mixing sequences of random variables. Stationarity is not assumed, but the variances of the partial sums must grow linearly. Our theorem extends previous results by supplying sufficient conditions for the weak convergence of the partial sum process to the Wiener process under less restrictive moment assumptions. For instance, if for some >0, and the mixing rate is exponential, then this functional c.l.t. holds. Under the weaker assumption with =0, the c.l.t. may fail to hold, and it is possible that the c.l.t. is satisfied, but the sequence of partial sum processes is not tight.     

Convergence of processes of step sums on mixing processes

We prove a limit theorem on the convergence of nonhomogeneous centered processes of step sums, constructed on random mixing sequences, to a process with independent increments. We also prove a invariance-principle type theorem for schemes of summation of functionals on random mixing sequences.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 716–721, May, 1990.     

Counter-examples to non-noetherian Elkik's approximation theorem

Elkik established a remarkable theorem that can be applied for any noetherian henselian ring. For algebraic equations with a formal solution (restricted by some smoothness assumption), this theorem provides a solution adically close to the formal one in the base ring. In this paper, we show that the theorem would fail for some non-noetherian henselian rings. These rings do not satisfy several conditions weaker than noetherianness, such as weak proregularity (due to Grothendieck et al.) of the defining ideal. We describe the resulting pathologies.     

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本文在{ξi}为强混合样本,{ani}是实三角阵列下,得到了一个新的关于线性和n∑i=1aniξi的中心极限定理.并利用该中心极限定理,进一步建立了线性过程部分和的中心极限定理.     

A general central limit theorem for strong mixing sequences

A central limit theorem for strong mixing sequences is given that applies to both non-stationary sequences and triangular array settings. The result improves on an earlier central limit theorem for this type of dependence given by Politis, Romano and Wolf in 1997.     

Local boundary controllability for the semilinear plate equation

weijiu_liu@uow.edu.

The problem of local controllability for the semilinear plate equation with Dirichlet boundary conditions is studied. By making use of Schauder's fixed point theorem and the inverse function theorem, we prove that this system is locally controllable under a super-linear assumption on the nonlinearity, that is, the initial states in a small neighborhood of 0 in a certain function space can be driven to rest by Dirichlet boundary controls. Our super-linear assumption includes the critical exponent.     

A stationary,pairwise independent,absolutely regular sequence for which the central limit theorem fails

Summary A strictly stationary finite-state non-degenerate random sequence is constructed which satisfies pairwise independence and absolute regularity but fails to satisfy a central limit theorem. The mixing rate for absolute regularity is only slightly slower than that in a corresponding central limit theorem of Ibragimov.This work was partially supported by NSF grant DMS 86-00399     

On strictly weakly mixing <Emphasis Type="Italic">C</Emphasis>*-dynamical systems

We consider strictly ergodic and strictly weakly mixing C*-dynamical cystems. We establish that a system is strictly weakly mixing if and only if its tensor product is strictly ergodic and strictly weakly mixing. We also investigate some weighted uniform ergodic theorem with respect to S-Besicovitch sequences for strictly weakly mixing dynamical systems.     

Neighborhood Turnpike Theorem for Continuous-Time Optimization Models

A neighborhood turnpike theorem is proved for continuous-time, infinite-horizon optimization models with positive discounting. Our approach is a primal one and no differentiability assumption is made. The basic hypothesis is a condition of uniform concavity at the point defining the undiscounted steady state. The main novelty here is that we formulate the theorem by taking the undiscounted steady state as the turnpike.     

A limit theorem for mixing processes subject to rarefaction. I

We prove a limit theorem on the approximation of point mixing processes subject to rarefaction by general renewal processes. This theorem contains a weaker condition on the mixing coefficient than the known conditions. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 471–475, April, 1998.     

Wolff–Denjoy theorems in nonsmooth convex domains

We give a short proof of Wolff–Denjoy theorem for (not necessarily smooth) strictly convex domains. With similar techniques we are also able to prove a Wolff–Denjoy theorem for weakly convex domains, again without any smoothness assumption on the boundary.     

Homotopy classification of equivariant regular sections

In his paper [2], Bierstone proves the equivariant Gromov theorem which is an integrability theorem for “open regularity condition” of equivariant sections of a smooth G-fibre bundle under the assumption that all orbit bundles of base manifold are non-closed. Here, we prove the result without his assumption under a nice “open regularity condition” which we call “G-extensible”. One of the examples of “G-extensible condition” is given by notions of Thom-Boardman singularities.     

A law of large numbers result for a bifurcating process with an infinite moving average representation

This paper derives a law of large numbers theorem for bifurcating processes defined on a perfect binary tree. This theorem can be viewed as a generalization of some results that have already appeared in the literature. For instance, all that is required of the bifurcating process is an infinite moving average representation with geometrically decaying coefficients and a finite moment assumption. In addition, the summands are assumed to belong to a flexible class of functions that satisfy a generalized Lipschitz type condition. These two criteria allow for an expansive range of applicability. Two examples are given as corollaries to the theorem.     

非凸向量集值优化Benson真有效解的最优性条件与对偶

在无需偏序锥内部非空的情况下给出了非凸约束向量集值优化Benaon真有效解一种加细的最优性条件，并建立了向量集值优化Benson真有效解一种改进的Lagrange乘子型对偶，它比已有的Lagrange乘子型对偶具有较好的对偶性。