Moments and tails in monotone-separable stochastic networks

强定律、随机单调向量场以及非正曲率度量空间上的梯度流

Strong laws, random monotone vector fields and gradient flows on metric spaces of nonpositive curvature

本文在Hadamard空间上推广了Sturm强大数定律，建立了随机单调向量场的强定律，并由此推导出积分函数梯度流的概率Lie-Trotter-Kato公式，同时首次给出次微分与积分可交换性的非线性版本。

Multivariate geometric distributions, (logarithmically) monotone sequences, and infinitely divisible laws (with erratum by Natalia Shenkman)

Two stochastic representations of multivariate geometric distributions are analyzed, both are obtained by lifting the lack-of-memory (LM) property of the univariate geometric law to the multivariate case. On the one hand, the narrow-sense multivariate geometric law can be considered a discrete equivalent of the well-studied Marshall-Olkin exponential law. On the other hand, the more general wide-sense geometric law is shown to be characterized by the LM property and can differ significantly from its continuous counterpart, e.g., by allowing for negative pairwise correlations. For both families…

On the stationary tail index of iterated random Lipschitz functions

Let $Ψ_1,Ψ_2,...$ be a sequence of i.i.d. random Lipschitz functions on a complete separable metric space with unbounded metric $d$ and forward iterations $X_n$. Suppose that $X_n$ has a stationary distribution. We study the stationary tail behavior of the functional $D_n=d(x_0,X_n)$, $x_0$ an arbitrary reference point, by providing bounds for these random variables in terms of simple contractive iterated function systems on the nonnegative halfline. Our results provide bounds for the lower and upper tail index of $D_n$ and will be illustrated by a number of popular examples including the AR(1…

Extended de Finetti theorems for boolean independence and monotone independence

We construct several new spaces of quantum sequences and their quantum families of maps in sense of Sołtan. Then, we introduce noncommutative distributional symmetries associated with these quantum maps and study simple relations between them. We will focus on studying two kinds of noncommutative distributional symmetries: monotone spreadability and boolean spreadability. We provide an example of a spreadable sequence of random variables for which the usual unilateral shift is an unbounded map. As a result, it is natural to study bilateral sequences of random objects, which are indexed by inte…

Asymptotic behaviour of first passage time distributions for subordinators

In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt in 1987. In the particular case of a subordinator in the domain of attraction of a stable distribution the results …

Edge universality of separable covariance matrices

In this paper, we prove the edge universality of largest eigenvalues for separable covariance matrices of the form $\mathcal Q :=A^{1/2}XBX^*A^{1/2}$. Here $X=(x_{ij})$ is an $n\times N$ random matrix with $x_{ij}=N^{-1/2}q_{ij}$, where $q_{ij}$ are $i.i.d.$ random variables with zero mean and unit variance, and $A$ and $B$ are respectively $n \times n$ and $N\times N$ deterministic non-negative definite symmetric (or Hermitian) matrices. We consider the high-dimensional case, i.e. ${n}/{N}\to d \in (0, \infty)$ as $N\to \infty$. Assuming $\mathbb E q_{ij}^3=0$ and some mild conditions on $A$ …

笨拙与粗心：非单调优惠券收集器中的稳态进入流量

Clumsy and Careless: Stationary-Entry Flux in Non-monotone Coupon Collectors

本文研究了三种非单调优惠券收集器模型，通过稳态进入视角分析其完成机制，证明了稳态进入流量与指数击中律的关系，并探讨了固定损失和后损失模型的极限行为。