Extreme exchangeable random order processes by positive definite functions on semigroups

On the Dovbysh-Sudakov representation result

We present a detailed proof of the Dovbysh-Sudakov representation for symmetric positive definite weakly exchangeable infinite random arrays, called Gram-de Finetti matrices, which is based on the representation result of Aldous and Hoover for arbitrary (not necessarily positive definite) symmetric weakly exchangeable arrays.

Harmonic Analysis of Symmetric Random Graphs

This note attempts to understand graph limits as defined by Lovasz and Szegedy (2006)} in terms of harmonic analysis on semigroups. This is done by representing probability distributions of random exchangeable graphs as mixtures of characters on the semigroup of unlabeled graphs with node-disjoint union, thereby providing an alternative derivation of de Finetti's theorem for random exchangeable graphs.

Stein 方法、Markov 过程与随机矩阵的线性特征值统计量

Stein's method, Markov processes, and linear eigenvalue statistics of random matrices

本文将 Stein 方法的无穷小可交换对方法与 Markov 半群理论结合，证明不变随机变量函数的多元正态逼近及 Wasserstein 界，并用于得到 Johansson 随机矩阵线性特征值统计量定理的收敛速率。

Finite Partially Exchangeable Laws are Signed Mixtures of Product Laws

Given a partition $\{I_1,\ldots,I_k\}$ of $\{1,\ldots,n\}$, let $(X_1,\ldots,X_n)$ be random vector with each $X_i$ taking values in an arbitrary measurable space $(S,\mathscr{S})$ such that their joint law is invariant under finite permutations of the indexes within each class $I_j$. Then, it is shown that this law has to be a signed mixture of independent laws and identically distributed within each class $I_j$. The representation is unique if and only if the set of these signed measures is weakly compact. We provide a necessary condition for the existence of a nonnegative directing measure.…

A connection between the Ghirlanda--Guerra identities and ultrametricity

We consider a symmetric positive definite weakly exchangeable infinite random matrix and show that, under the technical condition that its elements take a finite number of values, the Ghirlanda--Guerra identities imply ultrametricity.

Weak Similarity Orbit of (log)-Self-Similar Markov Semigroups on the Euclidean Space

We start by identifying a class of pseudo-differential operators, generated by the set of continuous negative definite functions, that are in the weak similarity (WS) orbit of the self-adjoint log-Bessel operator on the Euclidean space. These WS relations turn out to be useful to first characterize a core for each operator in this class, which enables us to show that they generate a class, denoted by $\mathscr{P}$, of non-self-adjoint $\mathcal{C}_0$-contraction positive semigroups. Up to a homeomorphism, $\mathscr{P}$ includes, as fundamental objects in probability theory, the family of self-…