Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRescaled weighted random balls models and stable self-similar random fields
418
0
0.0
(
0
)
Authors
Jean-Christophe Breton
Ask ChatGPT about the research
No Arabic abstract
We consider weighted random balls in $real^d$ distributed according to a random Poisson measure with heavy-tailed intensity and study the asymptotic behaviour of the total weight of some configurations in $real^d$. This procedure amounts to be very rich and several regimes appear in the limit, depending on the intensity of the balls, the zooming factor, the tail parameters of the radii and of the weights. Statistical properties of the limit fields are also evidenced, such as isotropy, self-similarity or dependence. One regime is of particular interest and yields $alpha$-stable stationary isotropic self-similar generalized random fields which recovers Takenaka fields, Telecom process or fractional Brownian motion.
rate research
Read More
We consider a collection of Euclidean random balls in ${Bbb R}^d$ generated by a determinantal point process inducing interaction into the balls. We study this model at a macros-copic level obtained by a zooming-out and three different regimes --Gaussian, Poissonian and stable-- are exhibited as in the Poissonian model without interaction. This shows that the macroscopic behaviour erases the interactions induced by the determinantal point process.
For each $n ge 1$, let $mathrm{d}^n=(d^{n}(i),1 le i le n)$ be a sequence of positive integers with even sum $sum_{i=1}^n d^n(i) ge 2n$. Let $(G_n,T_n,Gamma_n)$ be uniformly distributed over the set of simple graphs $G_n$ with degree sequence $mathrm{d}^n$, endowed with a spanning tree $T_n$ and rooted along an oriented edge $Gamma_n$ of $G_n$ which is not an edge of $T_n$. Under a finite variance assumption on degrees in $G_n$, we show that, after rescaling, $T_n$ converges in distribution to the Brownian continuum random tree as $n to infty$. Our main tool is a new version of Pitmans additive coalescent (https://doi.org/10.1006/jcta.1998.2919), which can be used to build both random trees with a fixed degree sequence, and random tree-weighted graphs with a fixed degree sequence. As an input to the proof, we also derive a Poisson approximation theorem for the number of loops and multiple edges in the superposition of a fixed graph and a random graph with a given degree sequence sampled according to the configuration model; we find this to be of independent interest.
In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight paths between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are i.i.d. exponential random variables.
We present a general framework for uncertainty quantification that is a mosaic of interconnected models. We define global first and second order structural and correlative sensitivity analyses for random counting measures acting on risk functionals of input-output maps. These are the ANOVA decomposition of the intensity measure and the decomposition of the random measure variance, each into subspaces. Orthogonal random measures furnish sensitivity distributions. We show that the random counting measure may be used to construct positive random fields, which admit decompositions of covariance and sensitivity indices and may be used to represent interacting particle systems. The first and second order global sensitivity analyses conveyed through random counting measures elucidate and integrate different notions of uncertainty quantification, and the global sensitivity analysis of random fields conveys the proportionate functional contributions to covariance. This framework complements others when used in conjunction with for instance algorithmic uncertainty and model selection uncertainty frameworks.
We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results have been recently established rigorously in mathematics, especially in the context of algebraic topology. In application to real-world systems, the reviewed models are typically used as null models, so that we take a statistical stance, emphasizing, where applicable, the entropic properties of the reviewed models. We also review a collection of phenomena and features observed in these models, and split the presented results into two classes: phase transitions and distributional limits. We conclude with an outline of interesting future research directions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
