An extension of the Vitali-Hahn-Saks Theorem

The Vitali-Hahn-Saks theorem on the absolute continuity of the setwise limit of a sequence of bounded measures is extended to allow unbounded measures and convergence of integrals of continuous functions vanishing at infinity.

     

Open problems in renewal, coupling and Palm theory

This note presents four sets of problems. The first suggests the possibility of a limit theory for null-recurrent renewal processes similar to the theory in the positive recurrent case. The second concerns exact coupling of random walks on the line with step-lengths that are neither discrete nor spread out. The third concerns the coupling characterization of setwise convergence of distributions of stochastic processes to a stationary limit. The fourth concerns characterizations of mass-stationarity, a concept formalizing the intuitive idea that the origin is a typical location in the mass of a random measure. Mass-stationarity is an intrinsic characterization of Palm versions with respect to stationary random measures.     

Gaussian random measures

A Gaussian random measure is a mean zero Gaussian process η(A), indexed by sets A in a σ-field, such that η(ΣA_i)=Ση(A_i), where ΣA_i indicates disjoint union and the series on the right is required to converge everywhere, so η is a random signed measure. (This is in contrast to so-called second order random measures, which only require quadratic mean convergence.) The covariance kernel of η is the signed bimeasure ν₀(A,B)=Eη(A)η(B). We give a characterization of those bimeasures which are covariance kernels of Gaussian random measures, and we show that every Gaussian random measure has an exponentially integrable total variation and is a.s. absolutely continuous with respect to a fixed finite measure on the state space.     

Decomposition of transformations

The decomposition theorem for transformations without any additivity assumptions is proved. This result is new even for image transformations. We also introduce a new class of transformations with some additivity assumpt ions which includes image transformations. Using the transformation from this class we generalize the Aarnes factorization theorem to representable deficient topological measures. We also establish the relationship between the decomposition of a representable deficient topological measure and the decomposition of the transformation mentioned above.     

General random perturbations of hyperbolic and expanding transformations

Small random perturbations of a general form of diffeomorphisms having hyperbolic invariant sets and expanding maps are considered. The convergence of invariant measures of perturbations to the Sinaî-Bowen-Ruelle measure in the case of a hyperbolic attractor and to the smooth invariant measure in the expanding case are proved. The convergence of corresponding entropy characteristics and the approximation of the topological pressure by means of perturbations is considered as well.     

Derived random measures

A derived random measure is constructed by integration of a random process with respect to a random measure independent of that process. Basic distributional properties, a continuity theorem, sample path properties, a strong law of large numbers, and a central limit theorem for derived random measures are established. Applications are given to compounding and thinning of point processes and the measure of a random set.     

广义可加Fuzzy测度的Radon—Nikodym定理

本文在[1,2]的基础上证明了广义可加fuzzy测度的Radon-Nikodym定理和Lebes-gue分解定理,从而完善了可加fuzzy测度的理论。     

Concentration of the first eigenfunction for a second order elliptic operator

We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold when the diffusion constant ε goes to zero. We assume that the first order term is given by a vector field b, whose recurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the normalized eigenfunctions concentrate on the recurrent sets of maximal dimension where the topological pressure [Y. Kifer, Principal eigenvalues, topological pressure and stochastic stability of equilibrium states, Israel J. Math. 70 (1990) (1) 1–47] is attained. On the cycles and torii, the limit measures are absolutely continuous with respect to the invariant probability measure on these sets. We have determined these limit measures, using a blow-up analysis. To cite this article: D. Holcman, I. Kupka, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On intensities of perturbed random measures on Hausdorff spaces

Abstract

This article studies classes of random measures on topological spaces perturbed by stochastic processes (a.k.a. modulated random measures). We render a rigorous construction of the stochastic integral of functions of two variables and showed that such an integral is a random measure. We establish a new Campbell-type formula that, along with a rigorous construction of modulation, leads to the intensity of a modulated random measure. Mathematical formalism of integral-driven random measures and their stochastic intensities find numerous applications in stochastic models, physics, astrophysics, and finance that we discuss throughout the article.     

广义可加Fuzzy测度

本文首先引入Fuzzy集类上（广义）可加Fuzzy测度的有关概念，然后给出Fuzzy集上关于可加Fuzzy测度的Fuaay积分及有关定理，最后在一定条件下给出广义可加Fuzzy测度的一系列分解定理。     

Variational dimension of random sequences and its application

A concept of variational dimension is introduced for a random sequence with stationary increments. In the Gaussian case, the variational dimension in the limit coincides with the Hausdorff dimension of a proper random process. Applications of the concept are illustrated by examples of neurological data and network traffic analysis.     

Martingale-coboundary representation for a class of random fields

It is known that under some conditions, a stationary random sequence admits a representation as a sum of two sequences: one of them is a martingale difference sequence, and another one is a so-called coboundary. Such a representation can be used for proving some limit theorems by means of the martingale approximation. A multivariate version of such a decomposition is presented in the paper for a class of random fields generated by several commuting, noninvertible, probability preserving transformations In this representation, summands of mixed type appear, which behave with respect to some group of directions of the parameter space as reversed rnultiparameter martingale differences (in the sense of one of several known definitions), while they look as coboundaries relative to other directions. Applications to limit theorems will be published elsewhere. Bibliography: 14 titles.     

紧拓扑半群上概率测度的简单半群及其支撑集

本文讨论一类拓扑半群上概率测度的极限性质．首先在紧半群上研究测度的简单半群和它的支撑集的相依关系;然后讨论测度的卷积幂ｕｎ收敛到Ｈａａｒｒ测度的充要条件．     

Ergodic properties for some non-expanding non-reversible systems

We study several properties of invariant measures obtained from preimages, for non-invertible maps on fractal sets which model non-reversible dynamical systems. We give two ways to describe the distribution of all preimages for endomorphisms which are not necessarily expanding on a basic set Λ. We give a topological dynamics condition which guarantees that the corresponding measures converge to a unique conformal ergodic borelian measure; this helps in estimating the unstable dimension a.e. with respect to this measure with the help of Lyapunov exponents. When there exist negative Lyapunov exponents of this limit measure, we study the conditional probabilities induced on the non-uniform local stable manifolds by the limit measure, and also its pointwise dimension on stable manifolds.     

ON THE STRUCTURES OF RANDOM MEASURE AND POINT PROCESS CONVOLUTION SEMIGROUPS

§0.ＩｎｔｒｏｄｕｃｔｉｏｎＫｅｎｄａｌｌ[1]ｆｏｕｎｄｅｄｔｈｅＤｅｌｐｈｉｃｓｅｍｉｇｒｏｕｐｔｈｅｏｒｙａｎｄｕｓｅｄｔｈｉｓｔｈｅｏｒｙｔｏｓｔｕｄｙｔｈｅｓｔｒｕｃｔｕｒｅｓｏｆｔｈｅｒｅｎｅｗａｌｓｅｑｕｅｎｃｅｓｅｍｉｇｒｏｕｐａｎｄｔｈｅｓｔａｎｄａｒｄｐ-ｆｕｎｃｔｉｏｎｓｅｍｉｇｒｏｕｐ.Ｄａｖｉｄｓｏｎ[2,3],ＲｕｚｓａａｎｄＳｚ啨ｋｅｌｙ[4]ａｎｄＨｅ[5,6]ｇｅｎｅｒａｌｉｚｅｄｔｈｅＤｅｌｐｈｉｃｓｅｍｉｇｒｏｕｐｔｈｅｏｒｙａｎｄｕｓｅｄｔｈｅｉｒｒｅｓｕｌｔｓｔｏｓｔｕｄ…     

Geometry of the vacant set left by random walk on random graphs,Wright's constants,and critical random graphs with prescribed degrees

We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by ?erný and Teixeira study by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.     

Limit theorem for associated random measures

In this paper, we investigate a limit theorem for a nonstatioaryd-parameter array of associated random variables applying the criterion of the tightness condition in Donsker,M[1951]. Our resuits imply an extension to the nonstatioary case of Convergence of Probability Measure of Billingsley.P [1968]. and analogous results for thed-dimensional associated random measure. These results are also applied to show a new limit theorem for Poisson cluster random measures.     

Limits of analytic vector measures

The article attempts to determine when a vector measure is the limit of a sequence of analytic vector measures in the sense of convergence in semivariation and when it is the limit of a sequence of such measures in variation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1133–1135, August, 1992.     

关于任意离散随机序列的一个强偏差定理

引用极限对数似然比的概念作为任意随机序列联合分布与其边缘分布"不相似性"的度量,构造几乎处处收敛的上鞅,讨论了任意离散随机序列的强偏差定理.     

On the Distribution Behaviour of Sequences

This paper presents results concerning those sets of finite Borel measures μ on a locally compact Hausdorff space X with countable topological base which can be represented as the set of limit distributions of some sequence. They arc characterized by being nonanpty, closed, connected and containing only measures μ with μ(X) = 1 (if X is compact) or 0 ≤ μ(X) ≤ 1 (if X is not compact). Any set with this properties can be obtained as the set of limit distributions of a sequence even by rearranging an arbitrarily given sequence which is dense in the sense that the set of accumulation points is the whole space X. The typical case (in the sense of Baire categories) is that a sequence takes as limit distributions all possible measures of this kind. This gives new aspects for the recent theory of maldistribukd sequences.