Khrystyna Buchak, Lyudmyla Sakhno
In the paper we study the models of time-changed Poisson and Skellam-type processes, where the role of time is played by compound Poisson-Gamma subordinators and their inverse (or first passage time) processes. We obtain explicitly the probability distributions of considered time-changed processes and discuss their properties.
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In the paper we consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators $G(N(t))$ and derive the expressions for their hitting times. We also study the time-changed Poisson processes where the role of time is played by the processes of the form $G(N(t)+at)$ and by the iteration of such processes.
Using Stein's method and the Malliavin calculus of variations, we derive explicit estimates for the Gamma approximation of functionals of a Poisson measure. In particular, conditions are presented under which the distribution of a sequence of multiple Wiener-Ito stochastic integrals with respect to a compensated Poisson measure converges to a Gamma distribution. As an illustration, we present a quantitative version and a non-central extension of a classical theorem by de Jong in the case of degenerate U-statistics of order two. Several multidimensional extensions, in particular allowing for mi…
It is shown that for a non-decreasing self-similar stochastic process $T$ with independent increments, the range of $T$ forms a Poisson point process with $σ$-finite intensity if and only if the one-dimensional distribution of $T(1)$ is of the gamma type. This follows from a general hold-jump description of such processes $T$, and implies the known result that the spacings between consecutive points of a scale invariant Poisson point process, with intensity $θx^{-1} dx$, are the points of another scale invariant Poisson point process with the same intensity.
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…
Gamma approximation and Poisson--Gaussian invariance principle on Poisson chaos
该研究在固定泊松混沌上推广伽马近似的相关估计，建立同核泊松与高斯多重积分的不变性原理，为泊松混沌的正态四矩定理提供新证明，并揭示Lindeberg条件的关键作用。
Let Pi and Gamma be homogeneous Poisson point processes on a fixed set of finite volume. We prove a necessary and sufficient condition on the two intensities for the existence of a coupling of Pi and Gamma such that Gamma is a deterministic function of Pi, and all points of Gamma are points of Pi. The condition exhibits a surprising lack of monotonicity. However, in the limit of large intensities, the coupling exists if and only if the expected number of points is at least one greater in Pi than in Gamma.
This paper deals with sequences of random variables belonging to a fixed chaos of order $q$ generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for $q=2$ and $q=4$. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and $U$-statistics o…
Approximate Modeling for Supercritical Galton-Watson Branching Processes with Compound Poisson-Gamma Distribution
本文研究了超临界分支过程在平均子代分布接近 1 的渐进行为下的种群大小分布，证明其可近似为复合泊松-伽马分布，并通过实验验证了该模型在电子倍增器等级联乘法过程中的有效性。
In this paper we deal with the generalized Gamma processes and their compositions. For the compositions of two or more than two generalized Gamma processes we give, when possible, the explicit law whereas, in the other cases the representations in terms of Fox's H-functions are given. We also study the connections between iteration and product of random processes by exploiting the properties of the generalized Gamma processes, such a study allows us to obtain some striking result about the compositions of the Cauchy processes or fractional Brownian motions. Furthermore, we find out the partial…
We show that the chaos representation of some Compound Poisson Type processes displays an underlying intrinsic combinatorial structure, partly independent of the chosen process. From the computational viewpoint, we solve the arising combinatorial complexity by means of the moments/cumulants duality for the laws of the corresponding processes, themselves measures on distributional spaces, and provide a combinatorial interpretation of the associated 'extended' Fock spaces. From the theoretical viewpoint, in the case of the Gamma measure, we trace back such complexity to its 'simplicial part', i.…
Changing time of simple continuous-time Markov counting processes by independent unit-rate Poisson processes results in Markov counting processes for which we provide closed-form transition rates via composition of trajectories and with which we construct novel, simpler infinitesimally over-dispersed processes.
Extended One-Liners for the Gamma, Poisson, and Binomial Distributions
本文提出仅用初等运算将三个独立均匀分布精确变换为Gamma、Poisson和Binomial随机数的方法，扩展了单行算法。
Splitting phenomenon in the Sathe-Selberg theorem, mod-Poisson convergence with auxiliary randomisation and universality of the Gamma factor
本文用辅助随机化揭示的独立性结构，解释多类 mod-Poisson 收敛模型中 Gamma 因子的普适来源，并为 Sathe–Selberg 情形引入 delta-Zeta 分布。
Asymptotic laws for regenerative compositions: gamma subordinators and the like
对于$\widetilde{\cal R} = 1 - \exp(- {\cal R})$，通过对从属子的闭值域${\cal R}$进行指数变换而得到的随机闭集，通过记录$n$均匀随机点簇被$\widetilde{\cal R}$的点分隔时的簇大小，定义了一般正整数$n$的再生合成.我们专注于当$\widetilde{\cal R}$是从伽玛从属派生的组成部分$K_n$的数量。我们证明了对数渐近的时刻和中心极限定理$K_n$和其他功能的组成，如单例，双例等的数量。这项研究补充了我们以前的工作时，这些功能的渐近尾的Lévy措施是经常变化的$0+$。
A law of the iterated logarithm for the number of blocks in regenerative compositions generated by gamma-like subordinators
本文研究由类伽马从属过程生成的可再生随机组合，证明了当样本量趋于无穷时组合中块数满足重对数律，并附带证明了布朗运动与正则变化函数卷积的重对数律。
Gamma, Gaussian and Poisson approximations for random sums using size-biased and generalized zero-biased couplings
本文利用零偏置耦合及其推广，为广义多项式模型下随机和的 Gaussian、Gamma 及 Poisson 近似建立了显式误差界，放宽了独立性假设，并在独立情形下与已知结果竞争。
We use a natural ordered extension of the Chinese Restaurant Process to grow a two-parameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford's sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford's trees. In general, the Markov branching trees induced by the two-parameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to es…
Distinguished properties of the gamma process and related topics
我们研究伽马过程的基本性质及其与各种主题的关系，如泊松-狄利克雷措施和稳定过程。我们证明了伽玛过程关于一大群线性变换的拟不变性。我们还表明，它是一个重整化的稳定过程的极限，并具有等价的西格玛有限措施（准勒贝格）具有重要的不变性。伽马过程的新性质可以应用于Poisson-Dirichlet测度。我们还强调了伽玛过程和布朗运动之间的深刻相似性。上述主题的连接，使更多的透明的一些老的和新的事实，稳定和伽玛过程，和泊松-狄利克雷措施。
On a Gibbs characterization of normalized generalized Gamma processes
我们证明了最近在Lijoi，Prünster and Walker（2007）中得到的归一化广义Gamma过程的Gibbs特征，也可以通过利用Pitman（2003）中指数倾斜Poisson-Kingman模型的特征得到。我们还提供了一个完成这一结果调查的存在性归一化随机措施诱导交换吉布斯分区类型$α\in (-\infty, 0]$。
The two-parameter Poisson-Dirichlet distribution is the law of a sequence of decreasing nonnegative random variables with total sum one. It can be constructed from stable and Gamma subordinators with the two-parameters, $α$ and $θ$, corresponding to the stable component and Gamma component respectively. The moderate deviation principles are established for the two-parameter Poisson-Dirichlet distribution and the corresponding homozygosity when $θ$ approaches infinity, and the large deviation principle is established for the two-parameter Poisson-Dirichlet distribution when both $α$ and $θ$ app…
A note on Bayesian nonparametric priors derived from exponentially tilted Poisson-Kingman models
我们从Poisson-Kingman模型（Pitman，2003）中导出了一类具有回火α-稳定混合分布的归一化广义Gamma过程。依赖于这种结构，可以表明，在贝叶斯非参数，结果的统计利益的数量下，这些先验，像类似的布莱克威尔-麦奎因预测规则或分布的数量不同的元素观察到的样本，出现作为皮特曼的结果的直接后果。
In this paper, we define a fractional negative binomial process (FNBP) by replacing the Poisson process by a fractional Poisson process (FPP) in the gamma subordinated form of the negative binomial process. First, it is shown that the one-dimensional distributions of the FPP are not infinitely divisible. The long-range dependence of the FNBP, the short-range dependence of its increments and the infinite divisibility of the FPP and the FNBP are investigated. Also, the space fractional Polya process (SFPP) is defined by replacing the rate parameter $λ$ by a gamma random variable in the definitio…
Poisson Process Partition Calculus with applications to Exchangeable models and Bayesian Nonparametrics
本文讨论了一种基于分块的Fubini演算在Poisson过程中的应用。该方法是一个放大的贝叶斯技术开发的Lo和翁伽马/狄利克雷过程。模型的应用程序被认为都属于一个不均匀的空间扩展的大小偏置的框架中使用的Perman，Pitman和Yor。在一些成果中;然后，为这些模型开发了一个基于显式划分的演算，其中还包括一系列重要的指数变化的测量公式。这些结果被应用到获得结果的Levy-Cox模型，身份有关的两参数泊松-狄利克雷过程和其他过程，概括的马尔可夫-克莱因对应，微积分扩展中立的权利过程，除其他事项外。
We consider a particular Cox process from a Bayesian viewpoint and show that the Bayes estimator of the intensity measure is the so-called Pólya sum kernel, which occurred recently in the context of the construction of the so-called Papangelou processes. More precisely, if the prior, the directing measure of the Cox process, is a Poisson-Gamma random measure, then the posterior is again a Poisson-Gamma random measure and the Bayes estimator of the intensity is the Pólya sum kernel. Moreover, we extend this result to doubly stochastic Poisson-Gamma priors and give conditions under which one can…
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。
Orthogonal decompositions for Lévy processes with an application to the gamma, Pacsal, and Meixner processes
众所周知，在所有具有独立增量的过程中，本质上只有布朗运动和泊松过程具有混沌表示性质。因此，一个自然的问题出现了：在一般Lévy过程的情况下，什么是CRP的适当模拟。这里至少有三种可能的方法。第一个，由于伊藤，使用的CRP的布朗运动和泊松过程，以及通过这些过程的一个莱维过程的表示。第二种方法，由于Nualart和Schoutens，包括在代表任何平方可积的随机变量作为一个多个随机积分的总和，相对于一个家庭的正交化中心功率跳跃过程。第三种方法以前从未应用于Lévy过程，它使用了关于定义在核型空间对偶上的概率测度的多项式正交化的思想。本文的主要目的是在黎曼流形上的一般（$\R$ -值）Lévy过程的情况下发展这三种方法，并且（更重要的是）理解这些方法之间的关系。我们将得到的结果应用到伽玛，帕斯卡，和Meixner过程，在这种情况下，相关的正交多项式的分析变得基本上更简单，更丰富，比一般情况下。
Extreme Distance Distributions of Poisson Voronoi Cells
针对齐次 Poisson Voronoi 镶嵌中生成点到顶点的最小、最大距离，本文用 $10^8$ 个单元仿真并比较多种分布拟合，证明 Generalized Gamma 最优并给出参数估计与算法。
Phase transitions of composition schemes: Mittag-Leffler and mixed Poisson distributions
本文研究临界生成函数复合格式，提出扩展模型并揭示三参数 Mittag-Leffler 分布主导的双重相变与局部极限定理。
Exact Scale--Shape Factorization of the Typical Poisson--Voronoi Cell Volume in Arbitrary Dimension
研究欧几里得空间中平稳泊松 - 沃罗诺伊镶嵌的典型胞体体积，通过推导精确尺度 - 形状分解，得出沃罗诺伊花体积性质，得到相关恒等式和界，简化下尾问题并推导渐近，模拟验证结果。
The generalized negative binomial distribution (GNB) is a new flexible family of discrete distributions that are mixed Poisson laws with the mixing generalized gamma (GG) distributions. This family of discrete distributions is very wide and embraces Poisson distributions, negative binomial distributions, Sichel distributions, Weibull--Poisson distributions and many other types of distributions supplying descriptive statistics with many flexible models. These distributions seem to be very promising for the statistical description of many real phenomena. GG distributions are widely applied in si…
The Poisson process is one of the simplest stochastic processes defined in continuous time, having interesting mathematical properties, leading, in many situations, to applications mathematically treatable. One of the limitations of the Poisson process is the rare events hypothesis; which is the hypothesis of unitary jumps within an infinitesimal window of time. Although that restriction may be avoided by the compound Poisson process, in most situations, we don't have a closed expression for the probability distribution of the increments of such processes, leaving us options such as working wi…
We offer a unified approach to the theory of concave majorants of random walks by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant. This leads to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave majorant, which is then used to find a complete description of the concave majorant for a walk of infinite length. In the case where subsets of increments may have the same arithmetic mean, we investigate three nested comp…
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