Finite and countable infinite products of Probabilistic Normed Spaces

随机赋范k-Means：概率度量空间中聚类的一次范式转变

Random Normed k-Means: A Paradigm-Shift in Clustering within Probabilistic Metric Spaces

本文提出首个在概率度量空间中运行的k-means变体RNKM，用距离分布函数替代传统距离度量，在多种数据集上验证其优于k-means++、fuzzy c-means等方法的聚类性能，尤其擅长识别非线性可分结构。

Completeness with respect to the probabilistic Pompeiu-Hausdorff metric

The aim of the present paper is to prove that the family of all closed nonempty subsets of a complete probabilistic metric space $L$ is complete with respect to the probabilistic Pompeiu-Hausdorff metric $H$. The same is true for the families of all closed bounded, respectively compact, nonempty subsets of $L$. If $L$ is a complete random normed space in the sense of Šerstnev, then the family of all nonempty closed convex subsets of $L$ is also complete with respect to $H$.

马祖尔-乌拉姆定理的概率泛化

A Probabilistic Generalization of the Mazur-Ulam Theorem

本文将马祖尔-乌拉姆定理泛化到概率空间，证明几乎处处保持距离的映射几乎处处与全局欧几里得等距映射一致。

闵可夫斯基赋范空间中的布朗运动

Brownian motion in Minkowski normed spaces

本文构造了闵可夫斯基赋范空间中的布朗运动，作为奇异McKean-Vlasov随机微分方程的解，并证明了路径唯一性，这是首次在芬斯勒空间中构造与非线性热方程相关的随机过程。

Invariant measures via inverse limits of finite structures

Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are invariant under all permutations of the underlying set that fix all constants. These measures are constructed from inverse limits of measures on certain finite structures. We use this construction to obtain invariant probability measures concentrated on the classes of countable models of certain first-order theories, including measures that do not assign positive …

From n+1-level atom chains to n-dimensional noises

In quantum physics, the state space of a countable chain of (n+1)-level atoms becomes, in the continuous field limit, a Fock space with multiplicity n. In a more functional analytic language, the continuous tensor product space over R of copies of the space C^{n+1} is the symmetric Fock space Gamma_s(L^2(R;C^n)). In this article we focus on the probabilistic interpretations of these facts. We show that they correspond to the approximation of the n-dimensional normal martingales by means of obtuse random walks, that is, extremal random walks in R^n whose jumps take exactly n+1 different values.…

The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations

We consider a Markov chain on the space of (countable) partitions of the interval [0,1], obtained first by size biased sampling twice (allowing repetitions) and then merging the parts (if the sampled parts are distinct) or splitting the part uniformly (if the same part was sampled twice). We prove a conjecture of Vershik stating that the Poisson-Dirichlet law with parameter theta=1 is the unique invariant distribution for this Markov chain. Our proof uses a combination of probabilistic, combinatoric, and representation-theoretic arguments.

Discrete approximation of the free Fock space

We prove that the free Fock space ${\F}(\R^+;\C)$, which is very commonly used in Free Probability Theory, is the continuous free product of copies of the space $\C^2$. We describe an explicit embedding and approximation of this continuous free product structure by means of a discrete-time approximation: the free toy Fock space, a countable free product of copies of $\C^2$. We show that the basic creation, annihilation and gauge operators of the free Fock space are also limits of elementary operators on the free toy Fock space. When applying these constructions and results to the probabilistic…