The Central Limit Theorem for Nonhomogeneous Markov Chains

In this paper, we prove a new central limit theorem for nonhomogeneous Markov chain by using the martingale central limit theorem under the condition of convergence of transition probability matrices for nonhomogeneous Markov chain in Cesaro sense, which can not be implied by Dobrushin's work.     

Individual ergodic theorem for unitary maps of random matrices

Using simple techniques of finite von Neumann algebras, we prove a limit theorem for random matrices.

     

马氏环境中可数马氏链的Poisson极限率

研究了马氏环境中的可数马氏链,主要证明了过程于小柱集上的回返次数是渐近地服从Poisson分布。为此,引入熵函数h,首先给出了马氏环境中马氏链的Shannon-Mc Millan-Breiman定理,还给出了一个非马氏过程Posson逼近的例子。当环境过程退化为一常数序列时,便得到可数马氏链的Poisson极限定理。这是有限马氏链Pitskel相应结果的拓广。     

Characteristic parameter sets and limits of circulant Hermitian polygon transformations

Polygon transformations based on taking the apices of similar triangles constructed on the sides of an initial polygon are analyzed as well as the limit polygons obtained by iteratively applying such transformations. In contrast to other approaches, this is done with respect to two construction parameters representing a base angle and an apex perpendicular subdivision ratio. Furthermore, a combined transformation leading to circulant Hermitian matrices is proposed, which eliminates the rotational effect of the basic transformation. A finite set of characteristic parameter subdomains is derived for which the sequence converges to specific eigenpolygons. Otherwise, limit polygons turn out to be linear combinations of up to three eigenpolygons. This leads to a full classification of circulant Hermitian similar triangles based polygon transformations and their limit polygons. As a byproduct classical results as Napoleon’s theorem and the Petr-Douglas-Neumann theorem can be easily deduced.     

An extension of the Kreiss stability theorem to families of matrices of unbounded order

The Kreiss matrix theorem asserts three necessary and sufficient conditions for a family of matrices of fixed finite order to be L₂-stable: a resolvent condition (R), a triangularization condition (S) and a Hermitian norm condition (H). We extend the Kreiss theorem to families of matrices of finite but unbounded order with the restriction that the degrees of the minimal polynomials of all matrices in the family are less than a fixed constant. For such matrix families, we show that (R) and (H) remain necessary and sufficient for L₂-stability, while (S) must be replaced by a somewhat stronger “block triangularization” condition (S′). This extended Kreiss theorem permits a corresponding extension of the Buchanan stability theorem.     

Criteria for the stability of the finiteness property and for the uniqueness of Barabanov norms

A set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product. The extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research. In this article, we give a condition on a finite irreducible set of matrices which guarantees that the finiteness property holds not only for that set, but also for all sufficiently nearby sets of equal cardinality. We also prove a theorem giving conditions under which the Barabanov norm associated to a finite irreducible set of matrices is unique up to multiplication by a scalar, and show that in certain cases these conditions are also persistent under small perturbations.     

The Classification of Complex Factor-Representations of the Three-Dimensional Heisenberg Group over a Countable Group Field of Finite Characteristic

We consider a field F that is a direct limit of an increasing chain of finite fields, and describe the Bratteli diagram, complex factor-representations, and projective moduli of the Heisenberg group of 3 × 3 upper-triangular matrices with elements from F. Bibliography: 3 titles.     

Bethe树和Cayley树上奇偶马尔可夫链场的强极限定理

本文介绍了N元Bethe树TB,N(N元Cayley树TC,N)上的奇偶马尔可夫链场的定义,并通过构造两个非负鞅证得了随机变量序列的强极限定理,应用此强极限定理获得了奇偶马尔可夫链场上的一个强极限定理,作为它的推论得到了状态和状态序偶出现频率的一类强极限定理及其估计,从而推广了关于N元Bethe树上马氏链场和二进树上奇偶马氏链场的部分强极限定理.     

Central Limit Theorems for Markov-Connected Random Variables

The paper provides some central limit theorems for triangular arrays of Markov-connected random variables. It is assumed the Markov chain to satisfy condition (D₁) which is an generalization of strong Doeblin's condition (D_o). One result represents a central limit theorem without the assumption of finite variances.     

有限变形的极限分析定理

本文建立了有限变形的极限分析的变分原理(定理1),证明了它与有限变形的极限分析的全套方程和条件等价.本文又证明了:根据此变分原理求得的有限变形的极限载荷乘子,介于有限变形的极限分析的上限定理和下限定理所分别给出的上限解和下限解之间.     

Global convergence of modified multiplicative updates for nonnegative matrix factorization

Nonnegative matrix factorization (NMF) is the problem of approximating a given nonnegative matrix by the product of two nonnegative matrices. The multiplicative updates proposed by Lee and Seung are widely used as efficient computational methods for NMF. However, the global convergence of these updates is not formally guaranteed because they are not defined for all pairs of nonnegative matrices. In this paper, we consider slightly modified versions of the original multiplicative updates and study their global convergence properties. The only difference between the modified updates and the original ones is that the former do not allow variables to take values less than a user-specified positive constant. Using Zangwill’s global convergence theorem, we prove that any sequence of solutions generated by either of those modified updates has at least one convergent subsequence and the limit of any convergent subsequence is a stationary point of the corresponding optimization problem. Furthermore, we propose algorithms based on the modified updates that always stop within a finite number of iterations.     

用凯里-克莱因参数计算载体姿态的新方法*

本文用凯里-克莱因参数的新矩阵来表示刚体连续几个有限转动的合成,导出这种矩阵乘法可易性的一般规则,并用此法简便地证明了刚体有限转动定理.文中所得结论简明、易记,对计算载体姿态具有实用价值.     

Semigroup of matrices acting on the max-plus projective space

We investigate the action of semigroups of d×d matrices with entries in the max-plus semifield on the max-plus projective space. Recall that semigroups generated by one element with projectively bounded image are projectively finite and thus contain idempotent elements.In terms of orbits, our main result states that the image of a minimal orbit by an idempotent element of the semigroup with minimal rank has at most d! elements. Moreover, each idempotent element with minimal rank maps at least one orbit onto a singleton.This allows us to deduce the central limit theorem for stochastic recurrent sequences driven by independent random matrices that take countably many values, as soon as the semigroup generated by the values contains an element with projectively bounded image.     

A Poisson limit theorem for a strongly ergodic non-homogeneous Markov chain

A strongly ergodic non-homogeneous Markov chain is considered in the paper. As an analog of the Poisson limit theorem for a homogeneous Markov chain recurring to small cylindrical sets, a Poisson limit theorem is given for the non-homogeneous Markov chain. Meanwhile, some interesting results about approximation independence and probabilities of small cylindrical sets are given.     

A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory

We use ergodic theory to prove a quantitative version of a theorem of M.A. Berger and Y. Wang, which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices. The proof rests on a structure theorem for continuous matrix cocycles over minimal homeomorphisms having the property that all forward products are uniformly bounded.     

Universality for the Conjugate Gradient and MINRES Algorithms on Sample Covariance Matrices

We present a probabilistic analysis of two Krylov subspace methods for solving linear systems. We prove a central limit theorem for norms of the residual vectors that are produced by the conjugate gradient and MINRES algorithms when applied to a wide class of sample covariance matrices satisfying some standard moment conditions. The proof involves establishing a four-moment theorem for the so-called spectral measure, implying, in particular, universality for the matrix produced by the Lanczos iteration. The central limit theorem then implies an almost-deterministic iteration count for the iterative methods in question. © 2022 Wiley Periodicals LLC.     

A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states

In this paper we consider bounded operators on infinite graphs, in particular Cayley graphs of amenable groups. The operators satisfy an equivariance condition which is formulated in terms of a colouring of the vertex set of the underlying graph. In this setting it is natural to expect that the integrated density of states (IDS), or spectral distribution function, exists. We show that it can be defined as the uniform limit of approximants associated to finite matrices. The proof is based on a Banach space valued ergodic theorem which even allows explicit convergence estimates. Our result applies to a variety of group structures and colouring types, in particular to periodic operators and percolation-type Hamiltonians.     

A CLT for renewal processes with a finite set of interarrival distributions

We prove a central limit theorem for a renewal process based on a sequence of independent non-negative interarrival times whose distributions are taken from a finite set. The result extends the classical central limit theorem obtained by Takács (1956).     

Functional central limit theorems and their associated large deviation principles for products of random matrices

Summary This paper establishes a functional central limit theorem for a product of random matrices. The sequence of matrices form a stationary process which is a -mixing. The individual matrices in the product become closer and closer to the identity matrix with longer and longer products. In addition, these perturbations from the identity matrix have mean zero. A large deviation principle for the limit process is proved.     

Weak ergodicity and products of random matrices

A representation for a weakly ergodic sequence of (nonstochastic) matrices allows products of nonnegative matrices which eventually become strictly positive to be expressed via products of some associated stochastic matrices and ratios of values of a certain function. This formula used in a random setup leads to a representation for the logarithm of a random matrix product. If the sequence of random matrices is in addition stationary then automatically almost all sequences are weakly ergodic, and the representation is expressed in terms of an one-dimensional stationary process. This permits properties of products of random matrices to be deduced from the latter. Second moment assumptions guarantee that central limit theorems and laws of the iterated logarithm hold for the random matrix products if and only if they hold for the corresponding stationary process. Finally, a central limit theorem for some classes of weakly dependent stationary random matrices is derived doing away with the restriction of boundedness of the ratios of colum entries assumed by previous studies. Extensions beyond stationarity are discussed.