The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let $\eta_t$ be a Poisson point process of intensity $t\geq 1$ on some state space \Y and $f$ be a non-negative symmetric function on \Y^k for some $k\geq 1$. Applying $f$ to all $k$-tuples of distinct points of $\eta_t$ generates a point process $\xi_t$ on the positive real-half axis. The scaling limit of $\xi_t$ as $t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the the $m$-th smallest point of $\xi_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener-It\^o chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen-Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as Poisson $k$-flats, Poisson random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry

Central limit theorems for the radial spanning tree

Consider a homogeneous Poisson point process in a compact convex set in $d$-dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length

Limit theory for the Gilbert graph

For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random geometric graph, is investigated as the intensity of the Poisson point process is increased and the distance parameter goes to zero. The asymptotic expectation and covariance structure of a class of length-power functionals are computed. Distributional limit theorems are derived that have a Gaussian, a stable or a compound Poisson limiting distribution. Finally, concentration inequalities are provided using a concentration inequality for the convex distance

Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry

A Poisson or a binomial process on an abstract state space and a symmetric function $f$ acting on $k$-tuples of its points are considered. They induce a point process on the target space of $f$. The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein's method for Poisson process approximation, a Glauber dynamics representation for the Poisson process and the Malliavin formalism. As applications of the main result, error bounds for approximations of U-statistics by Poisson, compound Poisson and stable random variables are derived, and examples from stochastic geometry are investigated.Comment: Published at http://dx.doi.org/10.1214/15-AOP1020 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Light pollution - extent, effects and approaches. TAB-Fokus

In addition to the intended effects, the increasing use of artificial outdoor lighting also entails a number of undesirable side effects referred to as light pollution. Artificial lighting can disturb the circadian rhythms of humans and animals, which are controlled by the change of day and night, and is suspected of being involved in the development of various diseases. Moreover, the increasing illumination of the night influences the natural behaviour of animals. Besides habitat changes, the consequences are ranging from changes in hunting or reproductive behaviour to the deadly attraction effect of light sources, e. g. for insects. However, the longterm consequences of these changes for entire populations, communities or landscapes are still poorly understood. Options for reducing light pollution exist both technologically and in terms of regulation and approval of lighting installations

Moments and central limit theorems for some multivariate Poisson functionals

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-It\^o integrals with respect to the compensated Poisson process. Second, a multivariate central limit theorem is shown for a vector whose components admit a finite chaos expansion of the type of a Poisson U-statistic. The approach is based on recent results of Peccati et al.\ combining Malliavin calculus and Stein's method, and also yields Berry-Esseen type bounds. As applications, moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process of $k$-dimensional flats in $\R^d$ are discussed

A novel fluorescent pH probe for expression in plants

BACKGROUND: The pH is an important parameter controlling many metabolic and signalling pathways in living cells. Recombinant fluorescent pH indicators (pHluorins) have come into vogue for monitoring cellular pH. They are derived from the most popular Aequorea victoria GFP (Av-GFP). Here, we present a novel fluorescent pH reporter protein from the orange seapen Ptilosarcus gurneyi (Pt-GFP) and compare its properties with pHluorins for expression and use in plants. RESULTS: pHluorins have a higher pH-sensitivity. However, Pt-GFP has a broader pH-responsiveness, an excellent dynamic ratio range and a better acid stability. We demonstrate how Pt-GFP expressing Arabidopsis thaliana report cytosolic pH-clamp and changes of cytosolic pH in the response to anoxia and salt-stress. CONCLUSION: Pt-GFP appears to be the better choice when used for in vivo-recording of cellular pH in plants