Gravitational allocation to Poisson points

多集合分配占据分布的极限理论：通过MAO范数实现的正态与泊松近似

Limit Theory of the Multi-set Allocation Occupancy (MAO) Distribution: Normal and Poisson Approximations via MAO Norm

本文通过MAO范数研究多集合分配占据分布的极限理论，推导出正态与泊松近似，并验证其在不同渐进行为下的有效性。

A note on large deviations for the stable marriage of Poisson and Lebesgue with random appetites

Let $Ξ\subset\mathbb R^d$ be a set of centers chosen according to a Poisson point process in $\mathbb R^d$. Let $ψ$ be an allocation of $\mathbb R^d$ to $Ξ$ in the sense of the Gale-Shapley marriage problem, with the additional feature that every center $ξ\inΞ$ has an appetite given by a nonnegative random variable $α$. Generalizing some previous results, we study large deviations for the distance of a typical point $x\in\mathbb R^d$ to its center $ψ(x)\inΞ$, subject to some restrictions on the moments of $α$.

Percolation for the stable marriage of Poisson and Lebesgue with random appetites

Let $Ξ$ be a set of centers chosen according to a Poisson point process in $\mathbb R^d$. Consider the allocation of $\mathbb R^d$ to $Ξ$ which is stable in the sense of the Gale-Shapley marriage problem, with the additional feature that every center $ξ\inΞ$ has a random appetite $αV$, where $α$ is a nonnegative scale constant and $V$ is a nonnegative random variable. Generalizing previous results by Freire, Popov and Vachkovskaia (\cite{FPV}), we show the absence of percolation when $α$ is small enough, depending on certain characteristics of the moment of $V$.

Percolation for the stable marriage of Poisson and Lebesgue

Let $Ξ$ be the set of points (we call the elements of $Ξ$ centers) of Poisson process in $\R^d$, $d\geq 2$, with unit intensity. Consider the allocation of $\R^d$ to $Ξ$ which is stable in the sense of Gale-Shapley marriage problem and in which each center claims a region of volume $α\leq 1$. We prove that there is no percolation in the set of claimed sites if $α$ is small enough, and that, for high dimensions, there is percolation in the set of claimed sites if $α<1$ is large enough.

Ergodicity of Poisson products and applications

In this paper we study the Poisson process over a $σ$-finite measure-space equipped with a measure preserving transformation or a group of measure preserving transformations. For a measure-preserving transformation $T$ acting on a $σ$-finite measure-space $X$, the Poisson suspension of $T$ is the associated probability preserving transformation $T_*$ which acts on realization of the Poisson process over $X$. We prove ergodicity of the Poisson-product $T\times T_*$ under the assumption that $T$ is ergodic and conservative. We then show, assuming ergodicity of $T\times T_*$, that it is impossibl…

Optimal transport from Lebesgue to Poisson

This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite. Moreover, we give precise conditions for the latter which demonstrate a sharp threshold at d=2. The cost will be defined in terms of an arbitrary increasing function of the distance. The coupling will be realized by means of a transport map ("allocation map") which assigns to each Poisson point a set ("cell") of Lebesgue measure 1. In the case of quadratic costs, all these cell…

泊松过程间最优尾部的因子匹配

A factor matching of optimal tail between Poisson processes

针对维度$d≥3$的欧几里得空间中两个独立单位强度泊松点过程，构造了等变可测的因子完美匹配，其配对距离尾部分布以$b\textrm{exp}(-cr^d)$快速衰减，证明依赖于泊松点过程的类似尾部分配规则和分数完美匹配转完美匹配的定理。

社区保险中的线性风险分担：在复利泊松模型中减少破产概率

Linear Risk Sharing in Community-Based Insurance: Ruin Reduction in the Compound Poisson Model

本文研究社区保险中索赔发生时的比例风险分担，通过比较单独运营与参与池化的破产概率，证明在共同尺度家族、完全分配和精算公平条件下，池化可降低个体破产概率，且通过指数和对数正态示例验证了比例分担规则对偿付能力的影响。

Inferring the intensity of Poisson processes at the limit of the detector sensitivity (with a case study on gravitational wave burst search)

We consider the issue of reporting the result of search experiment in the most unbiased and efficient way, i.e. in a way which allows an easy interpretation and combination of results and which do not depend on whether the experimenters believe or not to having found the searched-for effect. Since this work uses the language of Bayesian theory, to which most physicists are not used, we find that it could be useful to practitioners to have in a single paper a simple presentation of Bayesian inference, together with an example of application of it in search of rare processes.

On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

In this paper we propose the notion of continuous-time dynamic spectral risk-measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk-measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk-measures, which are obtained by iterating a given spectral risk-measure (such as Expected Shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in th…

Phase Transitions in Gravitational Allocation

Given a Poisson point process of unit masses (``stars'') in dimension d>=3, Newtonian gravity partitions space into domains of attraction (cells) of equal volume. In earlier work, we showed the diameters of these cells have exponential tails. Here we analyze the quantitative geometry of the cells and show that their large deviations occur at the stretched-exponential scale. More precisely, the probability that mass exp(-R^gamma) in a cell travels distance R decays like exp(-R^f_d(gamma)) where we identify the functions f_d exactly. These functions are piecewise smooth and the discontinuities o…