Yuri Bakhtin
We prove that under an easily verifiable set of conditions a sequence of associated random fields converges under rescaling to the Poisson Point Process and give a couple of examples.
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We obtain Stein approximation bounds for stochastic integrals with respect to a Poisson random measure over ${\Bbb R}^d$, $d\geq 2$. This approach relies on third cumulant Edgeworth-type expansions based on derivation operators defined by the Malliavin calculus for Poisson random measures. The use of third cumulants can exhibit faster convergence rates than the standard Berry-Esseen rate for some sequences of Poisson stochastic integrals.
In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes $N_α(t)$, $N_β(t)$, $t>0$, we show that $N_α(N_β(t)) \overset{\text{d}}{=} \sum_{j=1}^{N_β(t)} X_j$, where the $X_j$s are Poisson random variables. We present a series of similar cases, the most general of which is the one in which the outer process is Poisson and the inner one is a nonlinear fractional birth process. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns c…
We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method applied in [22] on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples of application include transformations conditioned by random sets such as the convex hull of a Poisson random measure.
There is given a characterization of the geometric distribution by the independence of linear forms with random coefficients. The result is a discrete analog of the corresponding theorem on exponential distribution. The property of linear statistics independence is also a characterization of Poisson law. Keywords: geometric distribution; exponential distribution; Poisson distribution; linear forms; random coefficients
Fractional Compound Poisson Random Fields on Plane
本文研究平面上的复合泊松随机场及其分数变体，推导分布、微分方程和弱收敛结果，并引入空间与时空分数泊松随机场，讨论性质并给出时间变换表示。
We construct a tree-based dependence structure for the representation of binomial, Poisson and Gaussian random vectors having a given covariance matrix, using sums of independent random variables. This construction allows us to characterize the supermodular ordering of such random vectors via the componentwise ordering of their covariance matrices. Our method relies on the representation of dependent components using binary trees on the discrete $d$-dimensional hypercube $C_d$, and on Möbius inversion techniques. In the case of Poisson random vectors this approach involves Lévy measures on $C_…
Skellam compound Poisson approximation to the sums of symmetric Markov dependent random variables
研究对称马尔可夫相依三点随机变量之和的近似问题，用Skellam随机变量近似该和，并在局部、总变差和Wasserstein度量下估计精度，利用测度卷积性质完成证明。
In this paper we study the properties of the Poisson random measure and the Poisson integral associated with a G-Levy process. We prove that a Poisson integral is a G-Levy process and give the conditions which ensure that a Poisson integral belongs to a good space of random variables. In particular, we study the relation between the quasi- continuity of an integrand and the quasi-continuity of the integral. Lastly, we apply the results to establish the pathwise decomposition of a G-Levy process into a generalized G-Brownian motion and a pure-jump G-Levy process and prove that both processes be…
Given a Poisson process on a bounded interval, its random geometric graph is the graph whose vertices are the points of the Poisson process and edges exist between two points if and only if their distance is less than a fixed given threshold. We compute explicitly the distribution of the number of connected components of this graph. The proof relies on inverting some Laplace transforms.
Stochastic Hall-magnetohydrodynamics equations on ${\mathbb{R}}^{3}$ with random forces expressed in terms of the time homogeneous Poisson random measures are considered. We prove the existence of a global martingale solution. The construction of a solution is based on the Fourier truncation method, stochastic compactness method and a version of the Skorokhod theorem for non-metric spaces adequate for Poisson type random fields.
Given a simple transient random walk $(S_n)_{n\geq 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(ξ(s))_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(ξ(S_n))_{n\geq 0}$. Under suitable conditions, we make explicit the extremal index and show that the point process of exceedances converges to a compound Poisson point process. We give two examples for which the cluster size distribution can be made explicit.
In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev $ζ$-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous R{é}nyi theorem) to a considerably wider class of random indices with mixed Poisson distributions including, e. g., those with the (generalized) negative binomial distribution.
In this paper we study finite velocity planar random motions with an infinite number of possible directions, where the number of changes of direction is randomized by means of an inhomogeneous fractional Poisson distribution. We first discuss the properties of the distributions of the generalized fractional inhomogeneous Poisson process. Then we show that the explicit probability law of the planar random motions where the number of changes of direction is governed by this fractional distribution can be obtained in terms of Mittag-Leffler functions. We also consider planar random motions with r…
We show that in a broad class of random counting measures one may identify only three that are rescaled versions of themselves when restricted to a subspace. These are Poisson, binomial and negative binomial random measures. We provide some simple examples of possible applications of such measures.
Multivariate Poisson approximation of joint subgraph counts in random graphs via size-biased couplings
通过Chen-Stein方法和大小偏倚耦合，在Wasserstein距离下得到随机图中联合子图计数及多元超几何分布的多元泊松近似，并给出显式收敛速率。
Given a set of independent Poisson random variables with common mean, we study the distribution of their maximum and obtain an accurate asymptotic formula to locate the most probable value of the maximum. We verify our analytic results with very precise numerical computations.
Consider a graph on randomly scattered points in an arbitrary space, with two points $x,y$ connected with probability $ϕ(x,y)$. Suppose the number of points is large but the mean number of isolated points is $O(1)$. We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson appro…
In this paper we use a Malliavin-Stein type method to investigate Poisson and normal approximations for the measurable functions of infinitely many independent random variables. We combine Stein's method with the difference operators in theory of concentration inequalities to obtain explicit bounds on Wasserstein, Kolmogorov and total variation distances. When restricted to the functions of a finite number of independent random variables, our method provides new bounds in the normal approximation. Meanwhile, our bounds in Poisson approximation are first to obtain explicitly.
Poisson Approximation to Weighted Sums of Locally Dependent Random Variables
该研究利用 Stein 方法分析局部相依随机变量加权和的 Poisson 逼近，给出优于已有结论的误差界，并通过数值比较验证其有效性。
We investigate the high-dimensional asymptotic distributional behavior of the $\mathbf{f}$-vector of a random Vietoris-Rips complex, that is generated over a stationary Poisson point process in $[-\frac{1}{2}, \frac{1}{2}]^d$ as the space dimension and the intensity tend to infinity while the radius parameter tends to zero simultaneously.
This paper deals with the quantitative normal approximation of non-linear functionals of Poisson random measures, where the quality is measured by the Kolmogorov distance. Combining Stein's method with the Malliavin calculus of variations on the Poisson space, we derive a bound, which is strictly smaller than what is available in the literature. This is applied to sequences of multiple integrals and sequences of Poisson functionals having a finite chaotic expansion. This leads to new Berry-Esseen bounds in de Jong's theorem for degenerate U-statistics. Moreover, geometric functionals of inters…
In this paper we study the existence phase transition of the random fractal ball model and the random fractal box model. We show that both of these are in the empty phase at the critical point of this phase transition.
Project a collection of points on the high-dimensional sphere onto a random direction. If most of the points are sufficiently far from one another in an appropriate sense, the projection is locally close in distribution to the Poisson point process.
The group of affine transformations with rational coefficients acts naturally on the real line, but also on the $p$-adic fields. The aim of this note is to show that, for random walks whose laws have a finite first moment, all these actions are necessary and sufficient to describe the Poisson boundary, which is in fact the product of all the fields that contract in mean.
We study the connection between PDEs and Lévy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenious-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in …
The paper studies the spectral properties of large Wigner, band and sample covariance random matrices with heavy tails of the marginal distributions of matrix entries.
A Poisson-Alekseev-Gröbner formula through Malliavin calculus for Poisson random integrals
本文为Poisson随机测度驱动的SDE建立了Alekseev-Gröbner公式，将全局误差表示为系数差，并通过新的Malliavin导数定义处理预期项，适用于随机连续流过程。
We characterize the subexponential densities on $(0,\infty)$ for compound Poisson distributions on $[0,\infty)$ with absolutely continuous Lévy measures. As a corollary, we show that the class of all subexponential probability density functions on $\mathbb R_+$ is closed under generalized convolution roots of compound Poisson sums. Moreover, we give an application to the subexponential density on $(0,\infty)$ for the distribution of the supremum of a random walk.
The random connection model is a random graph whose vertices are given by the points of a Poisson process and whose edges are obtained by randomly connecting pairs of Poisson points in a position dependent but independent way. We study first and second order properties of the numbers of components isomorphic to given finite connected graphs. For increasing observation windows in an Euclidean setting we prove qualitative multivariate and quantitative univariate central limit theorems for these component counts as well as a qualitative central limit theorem for the total number of finite compone…
Fractional Poisson Random Fields on $\mathbb{R}^2_+$
本文在正平面上定义分数阶泊松随机场，给出其时间变换表示，用Adomian分解求概率质量函数闭式解，并推广到高维及复合情形。
This paper focuses on the problem of modeling for small world effect on complex networks. Let's consider the supercritical Poisson continuous percolation on $d$-dimensional torus $T^d_n$ with volume $n^d$. By adding "long edges (short cuts)" randomly to the largest percolation cluster, we obtain a random graph $\mathscr G_n$. In the present paper, we first prove that the diameter of $\mathscr G_n$ grows at most polynomially fast in $\ln n$ and we call it the Poisson Geometry Small World. Secondly, we prove that the random walk on $\mathscr G_n$ possesses the rapid mixing property, namely, the …
Path-dependent Poisson random measures and stochastic integrals constructed from general point processes
本文为连续时间强化学习提出一种路径依赖的泊松随机测度扩展，从一般点过程出发定义Mesgaki随机测度，并构造其随机积分与Itô公式。
We consider a random walk among a Poisson system of moving traps on ${\mathbb Z}$. In earlier work [DGRS12], the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time $t$ in the annealed case and show that it is subdiffusive. As a by-product, we obtain an upper bound on the number of so-called thin points of a one-dimensional random walk, as well as a bound on the total volume of the holes in the random walk's range.
The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups $Γ$ endowed with a rich boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich. A geometrical method for constructing the strip as a subset of the lamplighter group starting with a smaller strip in the base group $Γ$ is developed. Then, this method is applied to several classes of base groups $Γ$: groups with infinitely many ends, hyperbolic groups in the sense of Gromov…
We determine, to within O(1), the expected minimal position at level n in certain branching random walks. The walks under consideration have displacement vector (v_1,v_2,...), where each v_j is the sum of j independent Exponential(1) random variables and the different v_i need not be independent. In particular, our analysis applies to the Poisson-Dirichlet branching random walk and to the Poisson-weighted infinite tree. As a corollary, we also determine the expected height of a random recursive tree to within O(1).
Concentration inequalities for subgraph counts in random geometric graphs built over Poisson point processes are proved. The estimates give upper bounds for the probabilities $\mathbb{P}(N\geq M +r)$ and $\mathbb{P}(N\leq M - r)$ where $M$ is either a median or the expectation of a subgraph count $N$. The bounds for the lower tail have a fast Gaussian decay and the bounds for the upper tail satisfy an optimality condition. A special feature of the presented inequalities is that the underlying Poisson process does not need to have finite intensity measure. The tail estimates for subgraph counts…
This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of Stein's equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes to provide a Poisson process approximation result in terms of a new metric called $d_π$, stronger than the total variation distance, defined as the supremum over all Wasserstein distances between random vectors obtained by evaluating the …
We consider a type of stochastic nonlinear beam equation driven by Lévy noise. By using a suitable Lyapunov function and applying the Khasminskii test we show the nonexplosion of the mild solutions. In addition, under some additional assumptions we prove the exponential stability of the solutions.
We want to compute the cumulative distribution function of a one-dimensional Poisson stochastic integral $I(\krnl) = \displaystyle \int_0^T \krnl(s) N(ds)$, where $N$ is a Poisson random measure with control measure $n$ and $\krnl$ is a suitable kernel function. We do so by combining a Kolmogorov-Feller equation with a finite-difference scheme. We provide the rate of convergence of our numerical scheme and illustrate our method on a number of examples. The software used to implement the procedure is available on demand and we demonstrate its use in the paper.
Poisson approximation of fixed-degree nodes in weighted random connection models
本文研究了高密度加权随机连接模型中固定度顶点的泊松近似问题，基于权重分布左尾的小权重分位数建立了一般性准则，并通过截断论证与Poisson点过程泛函的定量泊松近似结果进行证明。
We consider a stationary Poisson hyperplane process with given directional distribution and intensity in $d$-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body $K$ and consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containing $K$. We study how well these random polytopes approximate $K$ (measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties of $K$.
This article employs the relation between probabilities of two consecutive values of a Poisson random variable to derive conditions for the weak convergence of point processes to a Poisson process. As applications, we consider the starting points of k-runs in a sequence of Bernoulli random variables and point processes constructed using inradii and circumscribed radii of inhomogeneous Poisson-Voronoi tessellations.
In this paper we consider the Porous Medium Equation and establish a relationship between its Kompanets-Zel'dovich-Barenblatt solution $u(\xd,t), \xd\in \mathbb R^d,t>0$ and random flights. The time-rescaled version of $u(\xd,t)$ is the fundamental solution of the Euler-Poisson-Darboux equation which governs the distribution of random flights performed by a particle whose displacements have a Dirichlet probability distribution and choosing directions uniformly on a $d$-dimensional sphere (see, e.g., \cite{dgo}). We consider the space-fractional version of the Euler-Poisson-Darboux equation and…
In this paper we study the existence phase transition of scale invariant random fractal models. We determine the exact value of the critical point of this phase transition for all models satisfying some weak assumptions. In addition, we show that for a large subclass, the fractal model is in the empty phase at the critical point. This subclass of models includes the scale invariant Poisson Boolean model and the Brownian loop soup. In contrast to earlier results in the literature, we do not need to restrict our attention to random fractal models generated by open sets.
Upper bounds for the probabilities $\mathbb{P}(F\geq \mathbb{E} F + r)$ and $\mathbb{P}(F\leq \mathbb{E} F - r)$ are proved, where $F$ is a certain component count associated with a random geometric graph built over a Poisson point process on $\mathbb{R}^d$. The bounds for the upper tail decay exponentially, and the lower tail estimates even have a Gaussian decay. For the proof of the concentration inequalities, recently developed methods based on logarithmic Sobolev inequalities are used and enhanced. A particular advantage of this approach is that the resulting inequalities even apply in set…
Using Malliavin operators together with an interpolation technique inspired by Arratia, Goldstein and Gordon (1989), we prove a new inequality on the Poisson space, allowing one to measure the distance between the laws of a general random vector, and of a target random element composed of Gaussian and Poisson random variables. Several consequences are deduced from this result, in particular: (1) new abstract criteria for multidimensional stable convergence on the Poisson space, (2) a class of mixed limit theorems, involving both Poisson and Gaussian limits, (3) criteria for the asymptotic inde…
A conditional compound Poisson process approach to the sparse Erdős-Rényi random graphs: moderate deviations
提出条件复合Poisson过程表示稀疏Erdős-Rényi随机图的连通分量，揭示其相变与零程模型凝聚转变的联系，并在非临界状态下推导了中等偏差原理。
We prove a Poisson limit theorem in the total variation distance of functionals of a general Poisson point process using the Malliavin-Stein method. Our estimates only involve first and second order difference operators and are closely related to the corresponding bounds for the normal approximation in the Wasserstein distance by Last, Peccati and Schulte (2016). As an application of this Poisson limit theorem, we consider a stationary Poisson point process in $\mathbb{R}^d$ and connect any two points whenever their distance is less than or equal to a prescribed distance parameter. This constr…
Let $[\mathcal{P}]$ be the points of a Poisson process on $\mathbb{R}^d$ and $F$ a probability distribution with support on the non-negative integers. Models are formulated for generating translation invariant random graphs with vertex set $[\mathcal{P}]$ and iid vertex degrees with distribution $F$, and the length of the edges is analyzed. The main result is that finite mean for the total edge length per vertex is possible if and only if $F$ has finite moment of order $(d+1)/d$.
Plant differently colored points in the plane, then let random points ("Poisson rain") fall, and give each new point the color of the nearest existing point. Previous investigation and simulations strongly suggest that the colored regions converge (in some sense) to a random partition of the plane. We prove a weak version of this, showing that normalized empirical measures converge to Lebesgue measures on a random partition into measurable sets. Topological properties remain an open problem. In the course of the proof, which heavily exploits time-reversals, we encounter a novel self-similar pr…
On the Poisson approximation of random diagonal sums of Bernoulli matrices
本文利用 Stein-Chen 方法建立伯努利矩阵随机对角和与泊松分布之间全变差、Wasserstein 及局部距离的显式不等式，并改进经典泊松逼近定理 4.A 的部分界。
Continuing the analysis initiated in Lachiéze-Rey and Peccati (2011), we use contraction operators to study the normal approximation of random variables having the form of a U-statistic written on the points in the support of a random Poisson measure. Applications are provided: to boolean models, and coverage of random networks.
In Arguin & Tai (2018), the authors prove the convergence of the two-overlap distribution at low temperature for a randomized Riemann zeta function on the critical line. We extend their results to prove the Ghirlanda-Guerra identities. As a consequence, we find the joint law of the overlaps under the limiting mean Gibbs measure in terms of Poisson-Dirichlet variables. It is expected that we can adapt the approach to prove the same result for the Riemann zeta function itself.
We establish, under the Cramer exponential moment condition in a neighbourhood of zero, the Extended Large Deviation Principle for the Random Walk and the Compound Poisson processes in the metric space $\V$ of functions of finite variation on $[0,\infty)$ with the modified Borovkov metric $\r(f,g)= \r_\B(\hat{f},\hat{g}) $, where $ \hat f(t)= f(t)/(1+t)$, $t\in \R$, and $\r_\B$ is the Borovkov metric. LDP in this space is "more precise" than that with the usual metric of uniform convergence on compacts.
We establish new lower bounds for the normal approximation in the Wasserstein distance of random variables that are functionals of a Poisson measure. Our results generalize previous findings by Nourdin and Peccati (2012, 2015) and Biermé, Bonami, Nourdin and Peccati (2013), involving random variables living on a Gaussian space. Applications are given to optimal Berry-Esseen bounds for edge counting in random geometric graphs.
On Two Parameter Time-Changed Poisson Random Fields with Drifts
本文研究了二元 Lévy 过程与逆子ordinator的复合，推导了其色散与自相关矩阵，并引入了带漂移的双参数时变 Poisson 随机场及其分布的分数阶微分方程。
On rate of convergence to the Poisson law of the number of cycles in the generalized random graphs
本文证明了带幂律分布随机顶点权重的广义随机图中固定大小环数量分布与泊松分布在全变差距离下具有$O(1/\sqrt{n})$的收敛阶，基于Chen--Stein方法及随机变量平方和与和之比的性质，该性质可推广至其他渐近问题。
In this paper, we construct a new family of random series defined on $\R^D$, indexed by one scaling parameter and two Hurst-like exponents. The model is close to Takagi-Knopp functions, save for the fact that the underlying partitions of $\R^D$ are not the usual dyadic meshes but random Voronoi tessellations generated by Poisson point processes. This approach leads us to a continuous function whose random graph is shown to be fractal with explicit and equal box and Hausdorff dimensions. The proof of this main result is based on several new distributional properties of the Poisson-Voronoi tesse…
We study the $β$ analogue of the nonintersecting Poisson random walks. We derive a stochastic differential equation of the Stieltjes transform of the empirical measure process, which can be viewed as a dynamical version of the Nekrasov's equation in [7, Section 4]. We find that the empirical measure process converges weakly in the space of cádlág measure-valued processes to a deterministic process, characterized by the quantized free convolution, as introduced in [11]. For suitable initial data, we prove that the rescaled empirical measure process converges weakly in the space of distributions…
We study the normal approximation of functionals of Poisson measures having the form of a finite sum of multiple integrals. When the integrands are nonnegative, our results yield necessary and sufficient conditions for central limit theorems. These conditions can always be expressed in terms of contraction operators or, equivalently, fourth cumulants. Our findings are specifically tailored to deal with the normal approximation of the geometric $U$-statistics introduced by Reitzner and Schulte (2011). In particular, we shall provide a new analytic characterization of geometric random graphs who…
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