The generalized gamma distribution shows up in many problems related to engineering, hydrology as well as survival analysis. Earlier work has been done that estimated the deviation of the exponential and the Weibull distribution from Benford's Law. We give a mathematical explanation for the Benfordness of the generalized gamma distribution and present a measure for the deviation of the generalized gamma distribution from the Benford distribution.
A class of scale mixtures of gamma(k)-distributions that are generalized gamma convolutions
设$ k >0 $为整数，$ Y $为标准Gamma $(k)$分布随机变量。设$ X $是一个独立的正随机变量，其密度为$ k.$阶的双曲单调（HM），则$Y\cdot X$和$Y/X $都有广义伽玛卷积（GGC）分布.这一结果扩展了Roynette等人2009年的结果，Roynette等人治疗了$ k=1 $病例，但未使用HM概念。应用在漂移理论的扩散和理论的指数泛函的Lévy过程中提到。
Let $Y$ be a standard Gamma(k) distributed random variable, $k>0$, and let $X$ be an independent positive random variable. We prove that if $X$ has a hyperbolically monotone density of order $k$ ($HM_k$), then the distributions of $Y\cdot X$ and $Y/X$ are generalized gamma convolutions (GGC). This result extends results of Roynette et al. and Behme and Bondesson, who treated respectively the cases $k=1$ and $k$ an integer. We give a proof that covers all $k>0$ and gives explicit formulas for the relevant functions that extend those found by Behme and Bondesson in the integer case.
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$α\in (-\infty, 0]$。
We establish the one-to one bilateral interrelations between an asymptotic behavior for the tail of distributions for random variables and its great moments evaluation. Our results generalize the famous Richter's ones.
School of Mathematics and Statistics, Wuhan University of Technology（武汉理工大学数学与统计学院）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）
In this paper, we study some aspects on random analysis on the Léevy stochastic processes with margins following generalized hyperbolic distributions generated by gamma laws. In particular we study the boundedness of its total variations and the quadratic variations. Next we give an empirical construction that enables the graphical representation of the paths of such stochastic processes. Comparisons with the Brownian motions are considered.
The grey incomplete gamma distributions was established by one of the authors in a previous publication. In this article we use the Kondratiev characterization theorem to identify those via a suitable Laplace transform with holomorphic functions with suitable properties. We establish theorems for the integration and convergence of sequences of these distributions. As direct applications of these analytic tools we give the examples of Donsker's delta function, the local time, identify the time-derivative of the process as a suitable distribution and define the Gamma grey Ornstein-Uhlenbeck proc…
We derive sub-Riemannian Ricci curvature tensor for sub-Riemannian manifolds. We provide examples including the Heisenberg group, displacement group, and Martinet sub-Riemannian structure with arbitrary weighted volumes, in which we establish analytical bounds for sub-Riemannian curvature dimension bounds and log-Sobolev inequalities. {These bounds can be used to establish the entropy dissipation results for sub-Riemannian drift diffusion processes on a compact spatial domain, in term of $L_1$ distance.} Our derivation of Ricci curvature is based on generalized Gamma $z$ calculus and $z$--Boch…
Fix a space dimension $d\ge 2$, parameters $α> -1$ and $β\ge 1$, and let $γ_{d,α, β}$ be the probability measure of an isotropic random vector in $\mathbb{R}^d$ with density proportional to \begin{align*} ||x||^α\, \exp\left(-\frac{\|x\|^β}β\right), \qquad x\in \mathbb{R}^d. \end{align*} By $K_λ$, we denote the Generalized Gamma Polytope arising as the random convex hull of a Poisson point process in $\mathbb{R}^d$ with intensity measure $λγ_{d,α,β}$, $λ>0$. We establish that the scaling limit of the boundary of $K_λ$, as $λ\rightarrow \infty$, is given by a universal `festoon' of piecewise pa…
Consider a list of labeled objects that are organized in a heap. At each time, object $j$ is selected with probability $p_j$ and moved to the top of the heap. This procedure defines a Markov chain on the set of permutations which is referred to in the literature as Move-to-Front rule. The present contribution focuses on the stationary search cost, namely the position of the requested item in the heap when the Markov chain is in equilibrium. We consider the scenario where the number of objects is infinite and the probabilities $p_j$'s are defined as the normalization of the increments of a subo…
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…
We investigate the distributional properties of two generalized Ornstein-Uhlenbeck (OU) processes whose stationary distributions are the gamma law and the bilateral gamma law, respectively. The said distributions turn out to be related to the self-decomposable gamma and bilateral gamma laws, and their densities and characteristic functions are here given in closed-form. Algorithms for the exact generation of such processes are accordingly derived with the advantage of being significantly faster than those available in the literature and therefore suitable for real-time simulations.
