Jean-Baptiste Gouéré
Using the Palm measure notion, we prove the existence of the diffraction measure of all stationary and ergodic point processes. We get precise expressions of those measures in the case of specific processes : stochastic subsets of Z^d, sets obtained by the ``cut-and-project'' method.
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The present state of mathematical diffraction theory for systems with continuous spectral components is reviewed and extended. We begin with a discussion of various characteristic examples with singular or absolutely continuous diffraction, and then continue with a more general exposition of a systematic approach via stationary stochastic point processes. Here, the intensity measure of the Palm measure takes the role of the autocorrelation measure in the traditional approach. We furthermore introduce a `Palm-type' measure for general complex-valued random measures that are stationary and ergod…
Stochastic point processes relevant to the theory of long-range aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. The latter is based on the classical theory of point processes and the Palm distribution. Several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the Poisson summation formula for lattice Dirac combs.
The logarithmic derivative of a point process plays a key role in the general approach, due to the third author, to constructing diffusions preserving a given point process. In this paper we explicitly compute the logarithmic derivative for determinantal processes on $\mathbb{R}$ with integrable kernels, a large class that includes all the classical processes of random matrix theory as well as processes associated with de Branges spaces. The argument uses the quasi-invariance of our processes established by the first author.
We study Palm measures of determinantal point processes with $J$-Hermitian correlation kernels. A point process $\mathbb{P}$ on the punctured real line $\mathbb{R}^* = \mathbb{R}_{+} \sqcup \mathbb{R}_{-}$ is said to be $\textit{balanced rigid}$ if for any precompact subset $B\subset \mathbb{R}^*$, the $\textit{difference}$ between the numbers of particles of a configuration inside $B\cap \mathbb{R}_{+}$ and $B\cap \mathbb{R}_{-}$ is almost surely determined by the configuration outside $B$. The point process $\mathbb{P}$ is said to have the $\textit{balanced Palm equivalence property}$ if any…
This tutorial provides an introduction to Palm distributions for spatial point processes. Initially, in the context of finite point processes , we give an explicit definition of Palm distributions in terms of their density functions. Then we review Palm distributions in the general case. Finally we discuss some examples of Palm distributions for specific models and some applications.
A Palm hierarchy for determinantal point processes with the confluent hypergeometric kernel, the decomposing measures in the problem of harmonic analysis on the infinite-dimensional unitary group
本文证明无限维酉群调和分析中分解测度参数平移1等价于取无穷远约化Palm测度，通过正交多项式系综的有限维近似及Palm测度与标度极限的交换性完成证明。
Self-similar co-ascent processes and Palm calculus
本文研究自相似过程的重整化首次到达桥，推广了Brownian协同上升概念，并证明协同上升过程即为记录测度Palm分布下的过程，基于自相似性补充了平稳性方法。
For a determinantal point process induced by the reproducing kernel of the weighted Bergman space $A^2(U, ω)$ over a domain $U \subset \mathbb{C}^d$, we establish the mutual absolute continuity of reduced Palm measures of any order provided that the domain $U$ contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the $H^\infty(U)$-module structure of $A^2(U, ω)$. A corollary is the quasi-invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of $U$.
This paper establishes a remarkable result regarding Palmdistributions for a log Gaussian Cox process: the reduced Palmdistribution for a log Gaussian Cox process is itself a log Gaussian Coxprocess which only differs from the original log Gaussian Cox processin the intensity function. This new result is used to study functionalsummaries for log Gaussian Cox processes.
Palm distributions of superposed point processes for statistical inference
本文研究了叠加点过程的Palm分布，提出了最小对比估计和射线噪声Cox过程的高阶Palm分布推导方法，为统计推断提供了新的工具。
On the Palm distribution of superposition of point processes
本文研究两个独立点过程叠加后的 Palm 分布，证明其可表示为仅依赖各分量 Palm 分布与矩测度的简单混合，并推广到多点过程及高阶情形。
We study the mutual regularity properties of Palm measures of point processes, and establish that a key determining factor for these properties is the rigidity-tolerance behaviour of the point process in question (for those processes that exhibit such behaviour). Thereby, we extend the results of Osada-Shirai, Bufetov and Olshanski to new ensembles, particularly those that are devoid of any determinantal structure. These include the zeroes of the standard planar Gaussian analytic function and several others.
For a determinantal point process $X$ with a kernel $K$ whose spectrum is strictly less than one, Andr{é} Goldman has established a coupling to its reduced Palm process $X^u$ at a point $u$ with $K(u,u)>0$ so that almost surely $X^u$ is obtained by removing a finite number of points from $X$. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from $X$, where we specify the distribution of the difference $ξ_u:=X\setminus X^u$. This is used for discussing the degree of repulsiveness in DPPs in terms of $ξ_u$, including Gin…
Stein's method, Gaussian processes and Palm measures, with applications to queueing
本文提出一种基于Stein方法的通用框架，利用Palm测度耦合对点过程驱动的随机过程进行定量高斯逼近，并将其应用于重流量情形下GI/GI/∞队列的分析。
We study determinantal point processes on $\mathbb{C}$ induced by the reproducing kernels of generalized Fock spaces as well as those on the unit disc $\mathbb{D}$ induced by the reproducing kernels of generalized Bergman spaces. In the first case, we show that all reduced Palm measures $\textit{of the same order}$ are equivalent. The Radon-Nikodym derivatives are computed explicitly using regularized multiplicative functionals. We also show that these determinantal point processes are rigid in the sense of Ghosh and Peres, hence reduced Palm measures $\textit{of different orders}$ are singula…
This exposition explains the basic ideas of Stein's method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigration-death process and Palm theory. The latter approach also enables us to define local dependence of point processes [Chen and Xia (2004)] and use it to study Poisson process approximation for locally dependent point processes and for dependent superposition of point processes.
Explicit description of Christoffel deformations and Palm measures of the Plancherel measure, the $z$-measures and the Gamma process
本文引入离散正交多项式系综的Christoffel形变，将其推广至泊松化Plancherel测度、$z$-测度等点过程，得到相关Palm测度新公式，并导出Gamma过程的新形变。
Unimodular Random Measured Metric Spaces and Palm Theory on Them
本文定义了幺模随机测度度量空间作为多种离散与非离散概念的共同推广，证明了弱极限、随机游走等一般性结果，并将Palm理论推广至该空间以简化计算。
Using an intrinsic approach, we study some properties of random fields which appear as tail fields of regularly varying stationary random fields. The index set is allowed to be a general locally compact Hausdorff Abelian group $\mathbb{G}$. The values are taken in a measurable cone, equipped with a pseudo norm. We first discuss some Palm formulas for the exceedance random measure $ξ$ associated with a stationary (measurable) random field $Y=(Y_s)_{s\in \mathbb{G}}$. It is important to allow the underlying stationary measure to be $σ$-finite. Then we proceed to a random field (defined on a prob…
For a Pfaffian point process we show that its Palm measures, its normalised compositions with multiplicative functionals, and its conditional measures with respect to fixing the configuration in a bounded subset are Pfaffian point processes whose kernels we find explicitly.
We present new Poisson process approximation results for stabilizing functionals of Poisson and binomial point processes. These functionals are allowed to have an unbounded range of interaction and encompass many examples in stochastic geometry. Our bounds are derived for the Kantorovich-Rubinstein distance using the generator approach to Stein's method. We give different types of bounds for different point processes. While some of our bounds are given in terms of coupling of the point process with its Palm version, the others are in terms of the local dependence structure formalized via the n…
We define a probability measure preserving and r-discrete groupoid that is associated to every invariant point process on a locally compact and second countable group. This groupoid governs certain factor processes of the point process, in particular the existence of Cayley factor graphs. With this method we are able to show that point processes on amenable groups admit all (and only admit) Cayley factor graphs of amenable groups, and that the Poisson point process on groups with Kazhdan's Property (T) admits no Cayley factor graphs. This gives examples of pmp countable Borel equivalence relat…
A hierarchy of Palm measures for determinantal point processes with gamma kernels
该研究针对带伽马核的行列式点过程，探究参数平移的核链层级关系，证明相邻核的一维扰动性质与对应测度单点Palm分布的绝对连续性，并显式给出Radon-Nikodým导数的形式。
Palm problems arising in BAR approach and its applications
本文研究 BAR 方法中马尔可夫过程在 Palm 分布下的结构保持与计数时刻状态变化问题，提出框架并应用于重流量下有限等待空间单服务器队列的平稳分布极限推导。
A Palm Space Approach to Non-Linear Hawkes Processes
本文利用 Palm 空间方法研究非线性 Hawkes 过程，将 Hawkes 性质刻画为非负序列子空间上的算子，并在放宽的线性增长条件下证明平稳过程的存在性，同时分析了指数记忆函数的唯一平稳情形及多项式激活函数下点累积的尺度行为。
In the first part of this paper we consider a general stationary subcritical cluster model in $\mathbb{R}^d$. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein-Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE-solution in the special case of a Poisson driven random connection model.
The tail process $\boldsymbol{Y}=(Y_{\boldsymbol{i}})_{\boldsymbol{i}\in\mathbb{Z}^d}$ of a stationary regularly varying random field $\boldsymbol{X}=(X_{\boldsymbol{i}})_{\boldsymbol{i}\in\mathbb{Z}^d}$ represents the asymptotic local distribution of $\boldsymbol{X}$ as seen from its typical exceedance over a threshold $u$ as $u\to\infty$. Motivated by the standard Palm theory, we show that every tail process satisfies an invariance property called exceedance-stationarity and that this property, together with the polar decomposition of the tail process, characterizes the class of all tail pro…
Theory of point processes, in particular Palm calculus within the stationary framework, plays a fundamental role in the analysis of spatial stochastic models of wireless communication networks. Neveu's exchange formula, which connects the respective Palm distributions for two jointly stationary point processes, is known as one of the most important results in the Palm calculus. However, its use in the analysis of wireless networks seems to be limited so far and one reason for this may be that the formula in a well-known form is based upon the Voronoi tessellation. In this paper, we present an …
The framework of Stein's method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem \refimportantproposition) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Wasserstein pseudometric is introduced for measuring the accuracy of point process approximation. The pseudometric provides a generalization of many metrics used so far, including the t…
Markov Renewal Theory for Transfer Operators and Point Processes on the Line
本文研究了一维平稳点过程的配对相关函数指数衰减问题，通过马尔可夫条件、几何混合性及指数矩条件，结合经典再生技术与马尔可夫链的几何混合性，证明了结果，并应用于统计力学中的两个模型。
For general thinning procedures, its inverse operation, the condensing, is studied and a link to integration-by-parts formulas is established. This extends the recent results on that link for independent thinnings of point processes to general thinnings of finite point processes. In particular, the classical integration-by-parts formulas appear as the example of independent thinnings. Moreover, the representation of the splitting kernel of finite point processes in terms of its reduced Palm kernels is extended to general thinnings. This link is studied in the context of discrete random variabl…
The first aim is to construct generalizations of Polya type point process by applying a branching mechanism to these point processes. Conditions are given under which these point processes satisfy an integration by parts formula. Furthermore we compute their Palm kernels, which turn out to be superpositions of different point processes. Secondly we identify all point processes whose local characteristics agree with these of a fixed branching of a Polya type point process as mixtures of branchings of Polya type point process and show that in this case also they are characterized by an integrati…
This paper proves the consistency property for the regularized maximum likelihood estimators (MLEs) of multivariate Hawkes processes (MHPs). It also develops an alternating minimization type algorithm (AA-iPALM) to compute the MLEs with guaranteed global convergence to the set of stationary points. The performance of this AA-iPALM algorithm on both synthetic and real-world data shows that AA-iPALM consistently improves over iPALM and PALM. Moreover, AA-iPALM is able to identify the causality structure in rental listings on craigslist and herd behavior in the NASDAQ ITCH dataset.
Couplings and Poisson approximation for stabilizing functionals of determinantal point processes
本文利用Palm测度耦合和关联性质证明了行列式点过程稳定泛函的Poisson逼近，并得出Ginibre过程中大最近邻距离点过程在缩放后收敛于Poisson点过程及最大最近邻距离的缩放规律。
We propose Mecke-Palm formulas for multiple integrals with respect to a Poisson random measure interlaced with its intensity measure. We apply such formulas to multiple mixed Lévy systems of Lévy processes and obtain moment formulas for products of interlaced multiple Poisson integrals.
We develop a theory of insertion and deletion tolerance for point processes. A process is insertion-tolerant if adding a suitably chosen random point results in a point process that is absolutely continuous in law with respect to the original process. This condition and the related notion of deletion-tolerance are extensions of the so-called finite energy condition for discrete random processes. We prove several equivalent formulations of each condition, including versions involving Palm processes. Certain other seemingly natural variants of the conditions turn out not to be equivalent. We ill…
Point Processes and Gap Statistics for Translation Surfaces
该研究从点过程视角探究平移曲面上鞍连接方向的行为，定义相关方向与斜率点过程，证明几乎所有平移曲面上方向点过程经缩放后依分布收敛于斜率点过程，还得到间隙分布的极限性质及主层中斜率点过程收敛于泊松点过程的结论。
We study point processes that consist of certain centers of point tuples of an underlying Poisson process. Such processes arise in stochastic geometry in the study of exceedances of various functionals describing geometric properties of the Poisson process. We use a coupling of the point process with its Palm version to prove a general Poisson limit theorem. We then combine our general result with the theory of asymptotic shapes of large cells (Kendall's problem) in random mosaics and prove Poisson limit theorems for large cells (with respect to a general size functional) in the Poisson-Vorono…
In this paper we study the generating functionals of several random packing processes: the classical Matérn hard-core model; its extensions, the $k$-Matérn models and the $\infty$-Matérn model, which is an example of random sequential packing process. We first give a sufficient condition for the $\infty$-Matérn model to be well-defined (unlike the other two, the latter may not be well-defined on unbounded spaces). Then the generating functional of the resulting point process is given for each of the three models as the solution of a differential equation. Series representations and bounds on t…
The aim of the paper is to evaluate the miss probability of a Least Recently Used (LRU) cache, when it is offered a non-stationary request process given by a Poisson cluster point process. First, we construct a probability space using Palm theory, describing how to consider a tagged document with respect to the rest of the request process. This framework allows us to derive a general integral formula for the expected number of misses of the tagged document. Then, we consider the limit when the cache size and the arrival rate go to infinity proportionally, and use the integral formula to derive…
This paper defines the class of càdlàg functional marked point processes (CFMPPs). These are (spatio-temporal) point processes marked by random elements which take values in a càdlàg function space, i.e. the marks are given by càdlàg stochastic processes. We generalise notions of marked (spatio-temporal) point processes and indicate how this class, in a sensible way, connects the point process framework with the random fields framework. We also show how they can be used to construct a class of spatio-temporal Boolean models, how to construct different classes of these models by choosing specif…
We show that $\mathbb{P} ( \ell_X(0,T] \leq 1)=(c_X+o(1))T^{-(1-H)}$, where $\ell_X$ is the local time measure at $0$ of any recurrent $H$-self-similar real-valued process $X$ with stationary increments that admits a sufficiently regular local time and $c_X$ is some constant depending only on $X$. A special case is the Gaussian setting, i.e. when the underlying process is fractional Brownian motion, in which our result settles a conjecture by Molchan [Commun. Math. Phys. 205, 97-111 (1999)] who obtained the upper bound $1-H$ on the decay exponent of $\mathbb{P} ( \ell_X(0,T] \leq 1)$. Our appr…
Fluctuation theory for spectrally negative Lévy processes killed by additive functionals
本文研究了被一般加性函数控制的谱负莱维过程的波动恒等式，通过正共自然加性函数（PcNAFs）框架，证明了退出问题和残差测度等波动恒等式在经典情况下的结构保持，并用广义尺度函数表达，扩展了李和帕尔莫斯基、李和周的结果。
Consider a homogeneous Poisson point process of the Euclidean plane and its Voronoi tessellation. The present note discusses the properties of two stationary point processes associated with the latter and depending on a parameter $θ$. The first one is the set of points that belong to some one-dimensional facet of the Voronoi tessellation and are such that the angle with which they see the two nuclei defining the facet is $θ$. The main question of interest on this first point process is its intensity. The second point process is that of the intersections of the said tessellation with a straight…
Consider a symmetrical conflict relationship between the points of a point process. The Matérn type constructions provide a generic way of selecting a subset of this point process which is conflict-free. The simplest one consists in keeping only conflict-free points. There is however a wide class of Matérn type processes based on more elaborate selection rules and providing larger sets of selected points. The general idea being that if a point is discarded because of a given conflict, there is no need to discard other points with which it is also in conflict. The ultimate selection rule within…
Although the study of weak convergence of superpositions of point processes to the Poisson process dates back to the work of Grigelionis in 1963, it was only recently that Schuhmacher [Stochastic Process. Appl. 115 (2005) 1819--1837] obtained error bounds for the weak convergence. Schuhmacher considered dependent superposition, truncated the individual point processes to 0--1 point processes and then applied Stein's method to the latter. In this paper, we adopt a different approach to the problem by using Palm theory and Stein's method, thereby expressing the error bounds in terms of the mean …
Hyperuniformity in regular trees
该研究针对正则树中不变局部平方可积点过程的超均匀性展开，证明其无几何超均匀性，给出谱超均匀性与隐身性定义，并构造出数方差增长慢于体积的隐身不变随机格点轨道实例。
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