Infinite Random Geometric Graphs

We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability p ? (0, 1){p \in (0, 1)} if the metric distance between the vertices is below a given threshold. For certain choices of V as a countable dense set in \mathbbRⁿ{\mathbb{R}^n} equipped with the metric derived from the L _∞-norm, it is shown that with probability 1 such infinite random geometric graphs have a unique isomorphism type. The isomorphism type, which we call GR _n , is characterized by a geometric analogue of the existentially closed adjacency property, and we give a deterministic construction of GR _n . In contrast, we show that infinite random geometric graphs in \mathbbR²{\mathbb{R}^{2}} with the Euclidean metric are not necessarily isomorphic.     

Rare Events in Random Geometric Graphs

This work introduces and compares approaches for estimating rare-event probabilities related to the number of edges in the random geometric graph on a Poisson point process. In the one-dimensional setting, we derive closed-form expressions for a variety of conditional probabilities related to the number of edges in the random geometric graph and develop conditional Monte Carlo algorithms for estimating rare-event probabilities on this basis. We prove rigorously a reduction in variance when compared to the crude Monte Carlo estimators and illustrate the magnitude of the improvements in a simulation study. In higher dimensions, we use conditional Monte Carlo to remove the fluctuations in the estimator coming from the randomness in the Poisson number of nodes. Finally, building on conceptual insights from large-deviations theory, we illustrate that importance sampling using a Gibbsian point process can further substantially reduce the estimation variance.

     

几何随机变量的卷积

使用组合数学与概率论的方法研究了几何随机变量的卷积.     

Geometric Measures for Random Curved Mosaics of Rd

This paper deals with stationary random mosaics of R^d with general cell shapes. As geometric measures concentrated on the i-skeleton (i = 0, 1,…,d) the i-dimensional surface area (volume) measure and (i — 1) different curvature measures are chosen. The corresponding densities are calculated as well as for the mosaics and their superpositions in terms of mean cell parameters and mean cell numbers. This leads to various relations between the characteristic which are applied, in particular, to two- and three-dimensional tessellations. A comparison with known formulas for mosaics with convex cells in R² and R³ is given.     

On the Geometric Densities of Random Closed Sets

Abstract

In many applications it is of great importance to handle evolution equations about random closed sets of different (even though integer) Hausdorff dimensions, including local information about initial conditions and growth parameters. Following a standard approach in geometric measure theory such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are singular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper we suggest to cope with these difficulties by introducing random generalized densities (distributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. In this last one we analyze mean generalized densities, and relate them to densities of the expected values of the relevant measures. Many models of interest in material science and in biomedicine are based on time dependent random closed sets, as the ones describing the evolution of (possibly space and time inhomogeneous) growth processes; in such a situation, the Delta formalism provides a natural framework for deriving evolution equations for mean densities at all (integer) Hausdorff dimensions, in terms of the local relevant kinetic parameters of birth and growth. In this context connections with the concepts of hazard function, and spherical contact distribution function are offered.     

Heavy-Tailed Random Walks on Complexes of Half-Lines

We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is governed by an irreducible Markov transition matrix, with associated stationary distribution \(\mu _k\). If \(\chi _k\) is 1 for one-sided half-lines k and 1 / 2 for two-sided half-lines, and \(\alpha _k\) is the tail exponent of the jumps on half-line k, we show that the recurrence classification for the case where all \(\alpha _k \chi _k \in (0,1)\) is determined by the sign of \(\sum _k \mu _k \cot ( \chi _k \pi \alpha _k )\). In the case of two half-lines, the model fits naturally on \({{\mathbb {R}}}\) and is a version of the oscillating random walk of Kemperman. In that case, the cotangent criterion for recurrence becomes linear in \(\alpha _1\) and \(\alpha _2\); our general setting exhibits the essential nonlinearity in the cotangent criterion. For the general model, we also show existence and non-existence of polynomial moments of return times. Our moments results are sharp (and new) for several cases of the oscillating random walk; they are apparently even new for the case of a homogeneous random walk on \({{\mathbb {R}}}\) with symmetric increments of tail exponent \(\alpha \in (1,2)\).     

随机环境下的MTAR模型的几何遍历性

在本文中,提出了随机环境下的MTAR模型的非常返性及其确定的导出序列几何遍历的几个充分条件.     

关于随机变量序列几何平均的强极限定理

本文利用似然比作为一般连续型随机变量相对独立随机变量偏差的一种随机性度量 ,用矩变换构造几乎处处收敛的上鞅 ,结合分析方法 ,得到了一种乘积形式的强极限定理 .     

The Random Parameters AACD Models and Their Geometric Ergodicity

This paper proposes a new type of random parameters AACD (RPAACD)models, which extends the AACD model. Depending on the state of the price process, theRPAACD models seem to be a valuable alternative to existing approaches and have the betteroverall performance. We give the transition probability of the process. Moreover byemploying the transition probability, we obtain the probability properties of the RPACDmodel.     

The Spectrum of a Random Geometric Graph is Concentrated

Consider n points, x ₁,... , x _n , distributed uniformly in [0, 1] ^d . Form a graph by connecting two points x _i and x _j if . This gives a random geometric graph, , which is connected for appropriate r(n). We show that the spectral measure of the transition matrix of the simple random walk on is concentrated, and in fact converges to that of the graph on the deterministic grid.      

Topological Mean Value Relations for Random Cell Complexes

There are investigated stationary random q-dimensional topological cell complexes in ?^d, in particular, random tessellations. General relationships between the mean values of topological characteristics are derived. Then they are specified for the cases d = 2, 3, 4.     

Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

Let $\Delta _{n-1}$ denote the $(n-1)$ -dimensional simplex. Let $Y$ be a random $d$ -dimensional subcomplex of $\Delta _{n-1}$ obtained by starting with the full $(d-1)$ -dimensional skeleton of $\Delta _{n-1}$ and then adding each $d$ -simplex independently with probability $p=\frac{c}{n}$ . We compute an explicit constant $\gamma _d$ , with $\gamma _2 \simeq 2.45$ , $\gamma _3 \simeq 3.5$ , and $\gamma _d=\Theta (\log d)$ as $d \rightarrow \infty $ , so that for $c < \gamma _d$ such a random simplicial complex either collapses to a $(d-1)$ -dimensional subcomplex or it contains $\partial \Delta _{d+1}$ , the boundary of a $(d+1)$ -dimensional simplex. We conjecture this bound to be sharp. In addition, we show that there exists a constant $\gamma _d< c_d <d+1$ such that for any $c>c_d$ and a fixed field $\mathbb{F }$ , asymptotically almost surely $H_d(Y;\mathbb{F }) \ne 0$ .     

考虑旋转壳几何非线性时的随机响应研究

提出了一种统计线性化迭代法（IMSL）。利用这种方法,在形成非线性几何关系等效线性项的基础上,建立了非线性振动方程的等效刚度矩阵。通过求解方程,分析了几何非线性对旋转壳随机响应的影响。     

A Note on Geometric Embeddings of Simplicial Complexes in a Euclidean Space

In this note we prove that if a simplicial complex K can be embedded geometrically in R ^m , then a certain linear system of equations associated with K possesses a small integral solution. Received July 5, 1998, and in revised form May13, 1999.     

稳定随机变量序列几何加权和的Chover重对数律

设｛Ｘｎ,ｎ≥ ０｝是独立同分布的随机变量序列,其分布函数是一个对称的指数为 ａ（０＜ ａ＜ ２）的稳定分布·本文证明了依概率 １有 ｌｉｍ ｓｕｐβ－ｌ－｜（ ｌ－βα）１／α∑∞ ｎ＝０βｎＸｎ＝ｅｘｐ（１／α）·     

Geometric Ergodicity of Gibbs and Block Gibbs Samplers for a Hierarchical Random Effects Model

We consider fixed scan Gibbs and block Gibbs samplers for a Bayesian hierarchical random effects model with proper conjugate priors. A drift condition given in Meyn and Tweedie (1993, Chapter 15) is used to show that these Markov chains are geometrically ergodic. Showing that a Gibbs sampler is geometrically ergodic is the first step toward establishing central limit theorems, which can be used to approximate the error associated with Monte Carlo estimates of posterior quantities of interest. Thus, our results will be of practical interest to researchers using these Gibbs samplers for Bayesian data analysis.     

几何随机图大连通分支覆盖面积及其在传感器网络中的应用

随机网络中的大连通分支能体现一个网络的连通情况,是几何随机图研究的-个热点,具有重要的理论意义和应用价值.本文利用渗流理论,研究了几何随机图大连通分支覆盖面积所具有的性质,并将理论结果应用到大型无线传感器网络中,研究了无线传感器网络覆盖的性质.研究结果表明,对于节点服从泊松分布的大型无线传感器网络,其大连通分支覆盖区域大小与总区域大小的比值趋于-个常数,且并估计出了2维空间中没有被大连通分支所覆盖的连通区域(本文称为空洞)的大小.这些结果为衡量无线传感器网络性能提供了理论基础,对实际布网和网络优化等具有一定的指导意义.     

Discrete-Time Pricing and Optimal Exercise of American Perpetual Warrants in the Geometric Random Walk Model

An American option (or, warrant) is the right, but not the obligation, to purchase or sell an underlying equity at any time up to a predetermined expiration date for a predetermined amount. A perpetual American option differs from a plain American option in that it does not expire. In this study, we solve the optimal stopping problem of a perpetual American option (both call and put) in discrete time using linear programming duality. Under the assumption that the underlying stock price follows a discrete time and discrete state Markov process, namely a geometric random walk, we formulate the pricing problem as an infinite dimensional linear programming (LP) problem using the excessive-majorant property of the value function. This formulation allows us to solve complementary slackness conditions in closed-form, revealing an optimal stopping strategy which highlights the set of stock-prices where the option should be exercised. The analysis for the call option reveals that such a critical value exists only in some cases, depending on a combination of state-transition probabilities and the economic discount factor (i.e., the prevailing interest rate) whereas it ceases to be an issue for the put.     

Geometric Variational Crimes: Hilbert Complexes,Finite Element Exterior Calculus,and Problems on Hypersurfaces

A recent paper of Arnold, Falk, and Winther (Bull. Am. Math. Soc. 47:281–354, 2010) showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkin-like approach, one solves a variational problem on a finite-dimensional subcomplex. In a seemingly unrelated research direction, Dziuk (Lecture Notes in Math., vol. 1357, pp. 142–155, 1988) analyzed a class of nodal finite elements for the Laplace–Beltrami equation on smooth 2-surfaces approximated by a piecewise-linear triangulation; Demlow later extended this analysis (SIAM J. Numer. Anal. 47:805–827, 2009) to 3-surfaces, as well as to higher-order surface approximation. In this article, we bring these lines of research together, first developing a framework for the analysis of variational crimes in abstract Hilbert complexes, and then applying this abstract framework to the setting of finite element exterior calculus on hypersurfaces. Our framework extends the work of Arnold, Falk, and Winther to problems that violate their subcomplex assumption, allowing for the extension of finite element exterior calculus to approximate domains, most notably the Hodge–de Rham complex on approximate manifolds. As an application of the latter, we recover Dziuk’s and Demlow’s a priori estimates for 2- and 3-surfaces, demonstrating that surface finite element methods can be analyzed completely within this abstract framework. Moreover, our results generalize these earlier estimates dramatically, extending them from nodal finite elements for Laplace–Beltrami to mixed finite elements for the Hodge Laplacian, and from 2- and 3-dimensional hypersurfaces to those of arbitrary dimension. By developing this analytical framework using a combination of general tools from differential geometry and functional analysis, we are led to a more geometric analysis of surface finite element methods, whereby the main results become more transparent.