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Questions tagged [stochastic-geometry]

For questions related to stochastic geometry .It is the study of random spatial patterns. Stochastic geometry leads to modelling and analysis tools such as Monte Carlo methods.

In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting of random measures.

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Expected number of edges of the convex hull

Pick $n$ points at random in a bounded region according to a given probability distribution. As $n \to \infty$, what is the limiting distribution of the number of edges (or vertices) of the convex hull formed by these points? I am aware of results…
vallev
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Suggestions to start Statistical Manifolds

I am a computer science PhD student, and I need statistical manifolds theory for my work. I am currently reading Differential Geometry of Curves and Surface by Kristopher Tapp and Carmo. I plan to study Lee's smooth Manifold next. (My supervisor's…
Gammaformat
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Using Campbell's formula to find intensity measure $\Theta$

Problem Let $X_1, \ldots,X_n \sim K$ i.i.d random variables on $\mathbb{R}^d$, and let $$ \beta_{n,k} = \{ (x_1,\ldots,x_k) \in \{ X_1, \ldots,X_n\}^k \mid i \neq j, x_i \neq x_j \forall i,j \in \{1,\ldots,n\}\} $$ be the set of all $k$-tuples from…
Jacobiman
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Spatial point process generation

Let $\varepsilon = \{\varepsilon_i\}_{i\in\mathbb{R}^d}$ be a collection of independent standard Poisson point processes on $\mathbb{R}^d$. That is, $\varepsilon_i$ are functions from a probability space to a space of simple counting measures on…
Albert Paradek
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Voronoi Flower characterization by extremal points

The voronoi flower of a compact set $F\subseteq\mathbb{R}^d$ with respect to a point $x$ is given by $$\mathcal{F}_x(F):=\bigcup_{y\in F}B_{\|y-x\|}(y).$$ I need to show that $\mathcal{F}_x(F)=\mathcal{F}_x(\text{conv}(F))$. This is a result that is…
venom
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Selecting uniformly random points of the upper half plane model of hyperbolic geometry

How do I select a set of uniformly random points from some finite portion of the upper half plane model $$\mathbb{H}= \{ (x ,y) \mid y > 0; x, y \in \mathbb{R} \}$$ with the usual hyperbolic metric? Usually, in flat space, I can select $x,y$…
apg
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