We study here the topology of information on the space of probability measures over Polish spaces that was defined in Hellwig (1996). We show that under this topology, a convergent sequence of probability measures satisfying a conditional independence property converges to a measure that also satisfies the same conditional independence property. This also corrects the proof of a claim in Hellwig (1996, Lemma 4). Additionally, we determine sufficient conditions on the Polish spaces and the topology over measures spaces under which a convergent sequence of probability measures is also convergent in the topology of information.
This expository paper is intended for a short self-contained introduction to the theory of infinite convolutions of probability measures on Polish semigroups. We give the proofs of the Rees decomposition theorem of completely simple semigroups, the Ellis--Żelazko theorem, the convolution factorization theorem of convolution idempotents, and the convolution factorization theorem of cluster points of infinite convolutions.
The aim of this note is to show, by providing an elementary way to construct counter-examples, that the strong Feller and the joint (space-time) continuity for a semigroup of Markov kernels on a Polish space are not enough to ensure the existence of an associated càdlàg Markov process on the same space. One such simple counter-example is the Brownian semigroup on $\mathbb{R}$ restricted to $\mathbb{R}\setminus \{0\}$, for which it is shown that there is no associated càdlàg Markov process. Using the same idea and results from potential theory we then prove that the analogous result with càdlàg…
We present an alternative proof of Sanov's theorem for Polish spaces in the weak topology that follows via discretization arguments. We combine the simpler version of Sanov's Theorem for discrete finite spaces and well chosen finite discretizations of the Polish space. The main tool in our proof is an explicit control on the rate of convergence for the approximated measures.
In this article we extend the notion of metric measure spaces to so-called metric two-level measure spaces (m2m spaces): An m2m space $(X, r, ν)$ is a Polish metric space $(X, r)$ equipped with a two-level measure $ν\in \mathcal{M}_f(\mathcal{M}_f(X))$, i.e. a finite measure on the set of finite measures on $X$. We introduce a topology on the set of (equivalence classes of) m2m spaces induced by certain test functions (i.e. the initial topology with respect to these test functions) and show that this topology is Polish by providing a complete metric. The framework introduced in this article is…
We provide sufficient conditions for uniqueness of an invariant probability measure of a Markov kernel in terms of (generalized) couplings. Our main theorem generalizes previous results which require the state space to be Polish. We provide an example showing that uniqueness can fail if the state space is separable and metric (but not Polish) even though a coupling defined via a continuous and positive definite function exists.
一个有标记的度量测度空间（mmm-空间）是一个三元组（X，r，mu），其中（X，r）是一个完备且可分的度量空间，而mu是XxI上的一个概率测度，它是由可能的标记组成的波兰空间I。本文研究了在一定条件下的所有标记度量测度空间（等价类）的空间。它作为状态空间出现在马尔可夫过程的构造中，马尔可夫过程在随机图中取值，例如描述种群模型中随机演化的谱系结构的树值动态。本文给出了研究随机mmm-空间的依分布收敛所需的mmm-空间空间的拓扑性质。推广Greven，Pfaffelhuber and Winter，2009中引入的Gromov-弱拓扑的概念，我们定义了标记Gromov-弱拓扑，它将mmm-空间的集合转化为波兰空间.我们给出了随机mmm-空间的分布族的紧性的一个特征，并确定了一个收敛决定函数代数，称为多项式。
We prove a large deviation principle for a sequence of point processes defined by Gibbs probability measures on a Polish space. This is obtained as a consequence of a more general Laplace principle for the non-normalized Gibbs measures. We consider three main applications: Conditional Gibbs measures on compact spaces, Coulomb gases on compact Riemannian manifolds and the usual Gibbs measures in the Euclidean space. Finally, we study the generalization of Fekete points and prove a deterministic version of the Laplace principle known as $Γ$-convergence. The approach is partly inspired by the wor…
Let $π\in Π(μ,ν)$ be a coupling between two probability measures $μ$ and $ν$ on a Polish space. In this article we propose and study a class of nonparametric measures of association between $μ$ and $ν$, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between $ν$ and the disintegration $π_{x_1}$ of $π$ with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of c…
General extensions of an inequality due to Rogozin, concerning the essential supremum of a convolution of probability density functions on the real line, are obtained. While a weak version of the inequality is proved in the very general context of Polish $σ$-compact groups, particular attention is paid to the group \(\mathbb{R}^d\), where the result can combined with rearrangement inequalities for certain linear images for a strong generalization. As a consequence, we obtain a unification and sharpening of both the \(\infty\)-Renyi entropy power inequality for sums of independent random vector…
We study the continuity and the measurability of the solution to Schrödinger's functional equation, with respect to space, kernel and marginals, provided the space of all Borel probability measures is endowed with the weak topology. This is a continuation of our previous result where the space of all Borel probability measures was endowed with the strong topology. As an application, we construct a convex function of which the moment measure is a given probability measure, by the zero noise limit of a class of stochastic optimal transportation problems.
Distance matrices, random metrics and Urysohn space
本文引入了波兰（=完全可分度量）空间的一个universum--距离矩阵的凸锥，并研究了它的几何性质。恰好在这种universum意义下的一般Polish空间是P. S. Urysohn在20世纪定义的所谓Urysohn空间，一般度量三元组（=具有概率Borel测度的度量空间）也是具有非退化测度的Urysohn空间。本文证明了度量空间的完备不变量是关于集中在距离矩阵锥上的无限对称群的所谓矩阵分布不变和遍历测度。这定义了一类重要的新的随机矩阵和家庭的随机度量的自然。
In Athreya, Löhr, Winter (2016), an invariance principle is stated for a class of strong Markov processes on tree-like metric measure spaces. It is shown that if the underlying spaces converge Gromov vaguely, then the processes converge in the sense of finite dimensional distributions. Further, if the underlying spaces converge Gromov-Hausdorff vaguely, then the processes converge weakly in path space. In this paper we systematically introduce and study the Gromov-vague and the Gromov-Hausdorff-vague topology on the space of equivalence classes of metric boundedly finite measure spaces. The la…
We study properties of some popular topology on the space of Borel probabilities on a topological ambient space in this paper. We show that the two types of popular vague topology are equivalent to each other in case the ambient space is LCH. The two types of setwise topology induced from two equivalent descriptions of setwisely sequential convergence of probability measures are also equivalent to each other regardless of the topology on the ambient space. We give explicit conditions for the two types of vague topology and the two types of setwise topology to be separable or metrizable on the …
The paper introduces the notion of state for involutive bisemilattices, a variety which plays the role of algebraic counterpart of paraconsistent weak Kleene logic and whose elements are represented as Plonka sum of Boolean algebras. We investigate the relations between states over an involutive bisemilattice and probability measures over the (Boolean) algebras in the Plonka sum representation and, the direct limit of these algebras. Moreover, we study completition of involutive bisemilattices, as pseudometric spaces, and the topology induced by the pseudometric.
The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study the asymptotic stability of evolution systems of probability measures of time inhomogeneous transition operators for nonautonomous stochastic systems. A general theoretical result on this topic is established in a Polish space by establishing some sufficient conditions which can be verified in applications. Our abstract results are applied to a stochastic latt…
For a Tychonoff space $X$, the constructions $\hat P(X)$ and $P_τ(X)$ of the spaces of probability Radon measures and probability $τ$-smooth measures on $X$ are considered. It is proved that these constructions determine functors in the category of Tychonoff spaces, which extend the functor $P$ of probability measures in the category of compacta. In this part we investigate general topological properties of the spaces $\hat P(X)$ and $P_τ(X)$, as well as categorial properties of the functors $\hat P$ and $P_τ$.
给定两个可测空间$H$和$D$，它们具有可数生成的$σ$ -代数、$H$上的完美先验概率测度$P_H$和一个采样分布$S: H \rightarrow D$，则存在一个相应的推理映射$I: D \rightarrow H$，该映射直到一组测度零为止是唯一的.因此，给定数据测量$μ: 1 \rightarrow D$，可以计算后验概率$\widehat{P_H}= I \circ μ$。该过程是迭代的：对于每个更新的概率$P_H$，我们获得新的联合分布，其又产生新的推断图$I$，并且对于每个附加的测量重复该过程。主要结果使用了Faden的正则条件概率的存在定理，它比波兰空间的设置更具有普遍性。这种不太严格的设置，然后允许非平凡的决策规则（艾伦伯格-摩尔代数）有限（以及非有限）空间，并提供了一个共同的框架，决策理论和贝叶斯概率。
We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein distance. This monad is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces. We prove that this Kantorovich monad arises from a colimit construction on finite power-like constructions, which formalizes the intuition that probability measures are limit…
We show that, for the space of Borel probability measures on a Borel subset of a Polish metric space, the extreme points of the Prokhorov, Monge-Wasserstein and Kantorovich metric balls about a measure whose support has at most n points, consist of measures whose supports have at most n+2 points. Moreover, we use the Strassen and Kantorovich-Rubinstein duality theorems to develop representations of supersets of the extreme points based on linear programming, and then develop these representations towards the goal of their efficient computation.
A hypertree, or $\mathbb{Q}$-acyclic complex, is a higher-dimensional analogue of a tree. We study random $2$-dimensional hypertrees according to the determinantal measure suggested by Lyons. We are especially interested in their topological and geometric properties. We show that with high probability, a random $2$-dimensional hypertree $T$ is apsherical, i.e. that it has a contractible universal cover. We also show that with high probability the fundamental group $π_1(T)$ is hyperbolic and has cohomological dimension $2$.
