Tightness for a family of recursion equations

A tightness criterion for random fields, with application to the Ising model

We present a criterion for a family of random distributions to be tight in local Hölder and Besov spaces of possibly negative regularity. We then apply this criterion to the magnetization field of the two-dimensional Ising model at criticality, answering a question of Camia, Garban and Newman.

Stochastic order characterization of uniform integrability and tightness

We show that a family of random variables is uniformly integrable if and only if it is stochastically bounded in the increasing convex order by an integrable random variable. This result is complemented by proving analogous statements for the strong stochastic order and for power-integrable dominating random variables. Especially, we show that whenever a family of random variables is stochastically bounded by a p-integrable random variable for some p>1, there is no distinction between the strong order and the increasing convex order. These results also yield new characterizations of relative c…

Tightness of invariant distributions of a large-scale flexible service system under a priority discipline

We consider large-scale service systems with multiple customer classes and multiple server pools; interarrival and service times are exponentially distributed, and mean service times depend both on the customer class and server pool. It is assumed that the allowed activities (routing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a Leaf Activity Priority (LAP) policy, which assigns static priorities to the activities in the order of sequential "elimination" of the tree leaves. We consider the scaling lim…

混合复合分布的Panjer递归的扩展

Extensions of Panjer's recursion for mixed compound distributions

本文扩展Panjer递归至混合复合分布，通过随机化参数处理依赖，提出递归算法计算总索赔概率，并研究可交换索赔规模，附数值示例验证。

Unimodular Random Trees

We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We also prove that uniform integrability of the degree distribution of a family of finite graphs implies tightness of that family for local convergence, also known as random weak convergence.

Convergence to stable laws for a class of multidimensional stochastic recursions

We consider a Markov chain $\{X_n\}_{n=0}^\8$ on $\R^d$ defined by the stochastic recursion $X_{n}=M_n X_{n-1}+Q_n$, where $(Q_n,M_n)$ are i.i.d. random variables taking values in the affine group $H=\R^d\rtimes {\rm GL}(\R^d)$. Assume that $M_n$ takes values in the similarity group of $\R^d$, and the Markov chain has a unique stationary measure $ν$, which has unbounded support. We denote by $|M_n|$ the expansion coefficient of $M_n$ and we assume $\E |M|^\a=1$ for some positive $\a$. We show that the partial sums $S_n=\sum_{k=0}^n X_k$, properly normalized, converge to a normal law ($\a\ge 2$…

任意维度下对数相关高斯场指数度量的紧致性

Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension

研究\(\mathbb{R}^d\)（\(d\geq2\)）上类似刘维尔量子引力度量的自然近似方案，通过特定高斯随机函数序列构建随机度量族，证明在亚临界阶段经重标度后度量紧致，子序列极限诱导欧几里得拓扑，并列出诸多开放问题。

Tightness of supercritical Liouville first passage percolation

Liouville first passage percolation (LFPP) with parameter $ξ>0$ is the family of random distance functions $\{D_h^ε\}_{ε>0}$ on the plane obtained by integrating $e^{ξh_ε}$ along paths, where $h_ε$ for $ε>0$ is a smooth mollification of the planar Gaussian free field. Previous work by Ding-Dubédat-Dunlap-Falconet and Gwynne-Miller has shown that there is a critical value $ξ_{\mathrm{crit}} > 0$ such that for $ξ< ξ_{\mathrm{crit}}$, LFPP converges under appropriate re-scaling to a random metric on the plane which induces the same topology as the Euclidean metric (the so-called $γ$-\emph{Liouvil…

随机递归的变换与Aldous-Shields-Athreya级联及相关平均流方程分析中的临界现象

Transformation of Stochastic Recursions and Critical Phenomena in the Analysis of the Aldous-Shields-Athreya Cascade and Related Mean Flow Equations

本文扩展了树上的概率论以研究Aldous-Shields-Athreya级联中的临界现象，并利用新发展的概率框架，通过单参数变换族揭示非线性平均流方程解的非唯一性及其与随机爆炸、渗流等临界现象的对应关系。