Questions tagged [geometric-probability]

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4
votes
0answers
81 views

Upper bound $\tau_C := \int_{\|x\| \le 1}(vol(C \cap (x + C))/vol(C))dx$ for a convex body $C \subseteq \mathbb R^n$, by reducing to a ball

Let $C$ be a convex body in $\mathbb R^n$, i.e a bounded convex subset of $\mathbb R^n$ which has nonempty interior, and which is (A) open, or (B) closed (I'm not sure one makes more sense; choose the ...
3
votes
0answers
87 views

Probability that a Voronoi cell contains exactly k random points

Consider two independent point processes in the unit square $[0,1]^2$. The two point processes are identically independent and typically binomial/Poisson. One, say $\Phi^*$, is used to generate a ...
4
votes
5answers
243 views

Marginal density of uniform spherical distribution

Suppose that $X$ is distributed uniformly in the scaled $n$-sphere $\sqrt{n} \mathbf{S}^{n-1} \subset \mathbf{R}^n$. Then apparently the distribution of $(X_1, \dots, X_k)$, the first $k < n$ ...
-1
votes
0answers
43 views

Berry-Esseen unit ball

How can I show that sampling a random vector from a uniform distribution over a $d$-dimensional unit ball is similar to sampling a random vector from a uniform distribution over $d$-dimensional ...
1
vote
0answers
61 views

Dependence rank: what is the size of the largest subcollection of random variables which is statistically independent?

Let $X_1,\ldots,X_p$ be random variables on the same space. Define their dependence rank, denoted $rank(X_1,\ldots,X_p)$ as the largest nonnegative integer $k$ such that there is a subcollection of $k$...
2
votes
1answer
112 views

Lower bound for mutual inner products of N random unit vectors in $\mathbb{R}^n$, N > n

I have $N$ independent random unit vectors $\{v_i\}$ in $\mathbb{R}^n$, where N > n. I need a concentration inequality of the form $$\text{P}(|v_i \cdot v_j| > \epsilon \,\,\,\, \forall i, j = 1, \...
4
votes
0answers
56 views

Which orthant probabilities are the largest? (For a multivariate normal distribution)

I have a $k$-dimensional multivariate normal distribution $X∼N(0,\Sigma)$ with covariance matrix $\Sigma$. $\Sigma$ has two distinct eigenvalues, say $\lambda_1 > \lambda_2$, with orthogonal ...
0
votes
1answer
53 views

Tail bounds for the absolute difference of a coupled pair of sub-Gaussian random variables

Let $P$ and $Q$ are sub-Gaussian distributions on $\mathbb R$, and $(X,X')$ be a coupling of $P$ and $Q$, i.e $(X,X') \sim \pi$ for some distribution on $\mathbb R^2$ with marginals $P$ and $Q$. ...
0
votes
1answer
78 views

On bounding a certain discrepancy between probability distributions on the symmetric group

Disclaimer. This is a follow up to a question I asked and answered on SE https://math.stackexchange.com/q/3579311/168758. The question was about upper-bounds. Here I'm interested in lower bounds, and ...
2
votes
1answer
123 views

Volume computation using probabilistic approach

Let $\mathbb{S}^{d-1}=\{v\in\mathbb{R}^d:\|v\|_2=1\}$, namely $d-$dimensional sphere. It is well-known that if a random vector $X$ is distributed uniformly on $\mathbb{S}^{d-1}$, then there exists i.i....
4
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2answers
303 views

Definition of random measures

Introducing the notion of a random measure, textbooks usually start with a locally compact second countable Hausdorff space. Where does this requirement come from? I would like to have a motivation ...
1
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0answers
69 views

Strong data-processing inequality ? Upper bound on a certain modified total-variation metric

Let $\mathcal X=(\mathcal X,d)$ be a Polish space equipped with the Borel sigma-algebra. Let $p\ge 1$ and $P_1,P_2$ be probability distributions on $\mathcal X$ such that $\max_{k=1,2}\int d(x,x_0)^...
0
votes
0answers
93 views

Upper-bound on optimal transport distance between uniform distribution on balls in metric space

Context: I'm studying the so-called "Ollivier-Ricci curvature theory" (e.g see this ref). I'm particularly interested with how concentration of measure (in particular, in the sense of the Otto-Villani ...
0
votes
1answer
103 views

Compute lower bound on $\min_{E} \mathcal N(0,\sigma^2 I_n)(E)$ subject to $vol(E \cap H_n(r)) / vol(H_n(r)) \ge p$ where $H_n(r)$ is $n$-hemisphere

Let $n \ge 2$ be an integer, which may be assumed to be very large. For $r > 0$, consider the hemi-sphere $H_n(r) := S_n(r) \cap (\mathbb R^+ \times \mathbb R^{n-1})$, where $$ S_n(r):= \{x \in \...
2
votes
0answers
71 views

Metric ($f$-divergence) on space of probability measures that satisfies pythagorean theorem

Let $E$ be a polish space, $\mathcal{P}(E)$ the Borel probability measures on $E$ with the topology of weak convergence and $\mathcal{Q} \subset \mathcal{P}(E)$ a convex and compact set. First, the ...
2
votes
1answer
246 views

On 4 random points in a rectangle [closed]

Given a bounded rectangular area, I generate 4 random points. What is the probability that the fourth point lie within a triangle formed the first 3? How would I attack this problem? The goal is to ...
3
votes
3answers
303 views

Introduction to information geometry and/or geometric control theory

Some background: I'vebeen searching for a research project to work through my grad studies and I found information geometry like a strong candidate but the amount of work out there is overwhelming. I ...
2
votes
0answers
126 views

Is there any geometric interpretation for the trace of Fisher information matrix?

Consider a parametric family $p_\theta(x)$ of distributions, with parameter $\theta \in \Theta \subseteq \mathbb R^p$. If the mapping $\theta \mapsto p_\theta(x)$ is continuously differentiable at $\...
5
votes
1answer
442 views

Show that the Markov chain of random tiling is irreducible

Consider a Markov chain on a state-space which is slightly weird: It is the space of all tilings of a hexagon as shown in the left-hand side of the figure below with three types of rhombi: yellow, ...
0
votes
1answer
55 views

Overlap count of convex combination of points [closed]

Let $x_1, ..., x_n\in\mathbb{R}^d$. We know that a point $x$ in the convex hull $\text{conv}(x_1, ..., x_n)$ may be expressed as convex combinations $x=\sum_{i=1}^n r_ix_i=\sum_{i=1}^n s_ix_i$ with ...
1
vote
0answers
87 views

Sampling uniformly from a ball in SPD manifold

I'm trying to sample uniformly from a ball around the identity matrix in the manifold of symmetric positive-definite matrices (SPD), i.e., $\mathcal{B}(R) = \{X \in \mathcal{P}(n) : d_\mathcal{P}(I_n, ...
2
votes
1answer
155 views

Uniform sampling on a Riemannian manifold via tangent space and exponential map

Given a Riemannian manifold $(\mathcal{M}, \{g_x\}_{x \in \mathcal{M}})$ and a fixed point $x \in \mathcal{M}$, does the following procedure yield uniform samples from $\{y \in \mathcal{M} : d_\...
2
votes
0answers
114 views

Intrinsic volume - is there a simplified formula?

I'm struggling to understand how can I compute the instrinsic volumes (or Minkowski functionals) of a submanifold $\Omega$ of $\mathbb{R}^N$. I found a formula, but I really can't understand it... It ...
2
votes
1answer
106 views

Barycenter Map on Wasserstein Space

Let $(X,d)$ be a complete separable metric space, $P_1(X,d)$ be the set of Radon probability measures on $X$ satisfying $$ P_1(X,d)\triangleq \left\{ \nu:\,(\exists x_0\in X)\, \int_{x\in X} d(x,x_0)d\...
1
vote
2answers
89 views

Minkowski functionals (valuations)

What can we say about a polyconvex set $\omega\subset\mathbb{R}^N$ knowing the values of its Minkowski valuations? I know that Hadwiger characterization theorem states that there are only $N+1$ ...
2
votes
0answers
252 views

Normal multivariate orthant probabilities

(Previously I posted a similar question on math.SE, hoping that this question would have an easy answer. As the question appears hard, I am hoping I can perhaps get more feedback here.) Let $\mathbf{...
0
votes
0answers
32 views

Determining Sample Spaces for Sylvester's Four Point Problem

I don't have any background in geometric probability, so may I ask for forgiveness if the following is wrong or doesn't make sense: Assumptions: no three sampled points are collinear the (naive) ...
2
votes
1answer
152 views

Isoperimetry on $[0, 1]^n$ w.r.t $\ell_p$ distance, with $p \in [1,\infty]$

Let $A$ be a measurable subset of the metric space $\mathcal X = ([0, 1]^n,\ell_p)$ with $1 \le p \le \infty$, and define its $\varepsilon$-blowup by $A^\varepsilon:=\{x \in \mathcal X \mid \|x-a\|_p \...
4
votes
2answers
143 views

Probabilities of four points being in convex/deltoid configurations

Question: what is the probability that four distinct points in general position in the Euclidean plane are in convex configuration, depending on the number of leaf nodes in their Minimum ...
2
votes
0answers
47 views

Given a set of marginals, what is the largest support of a distribution satisfying these?

Given a random variable $X$ with support over $\{0,1\}^I$, we can define the marginal distribution on the bits indexed by $A \subseteq I$ by $Pr(X_A = x_A) = \sum_{x \in \{0,1\}^{I - A}} Pr(X = x \cup ...
3
votes
2answers
230 views

What happens to the Gaussian volume of a Borel set when it is translated?

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$, $A \subseteq \mathbb R^n$ be Borel and $c \in \mathbb R^n$. Define the translate $A_c := c + A := \{c+a \mid a \in A\} = \{x \in \...
0
votes
0answers
58 views

Function classes with high Rademacher complexity

My question is two fold, Is there any general understanding of what makes a function class have high Rademacher complexity? (Sudakov minoration would say that one sufficient condition for a class of ...
1
vote
1answer
149 views

Unconditional lower bound for volume of blowup $\mu(B^\epsilon)$ for $\mu(B) \in (0, 1)$ and $\epsilon > 0$ not “too large”

For a Borel subset $B$ of a metric space $X = (X,d)$ and $\epsilon>0$, recall the defintion of the $\epsilon$-blowup of $B$, namely $B^\epsilon = \{x \in X | d(x,B) \le \epsilon\}$. Let $\mu$ be a ...
4
votes
2answers
116 views

Complexity of Random Delaunay Triangulation in 3D

My question: Is the number of cells in a three-dimensional Poisson-Delaunay triangulation with $n$ vertices $\mathcal O(n)$ with probability one? which is equivalent to the question Is the ...
10
votes
3answers
431 views

Mean maximum distance for N random points on a unit square

Following up on Mean minimum distance for N random points on a one-dimensional line and Mean minimum distance for N random points on a unit square (plane), I have the following questions. Given N ...
2
votes
0answers
88 views

Inclusion of convex polytopes and embedding from $\ell_2$ to $\ell_\infty$

I would like to dig deeper into the problem posted Probability that a convex shape contains the unit ball: If you pick n points uniformly at random from the surface of a d dimensional sphere of ...
0
votes
3answers
499 views

Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
1
vote
1answer
77 views

gaussian isoperimetric result for minimal measure under translation

Consider two spherical Gaussian distributions in $\mathbb{R}^n$, $A = \mathcal{N}(x, I)$ and $B=\mathcal{N}(y, I)$ where the difference in means is $\delta = y - x$. Let $S \subset \mathbb{R}^n$ be a ...
1
vote
0answers
379 views

Closed-form formula for Wasserstein distance between uniform discrete distribution and discrete distribution with same support

Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{...
0
votes
1answer
141 views

Does log-concave approximable distribution satisfy transportation-cost inequality?

Definition: Recall that a distribution $\mu$ on $\mathbb R^d$ is said to be log-convave with constant $c > 0$, if density $d\nu \propto e^{-V}dvol$ satisfying the curvature condition $$ \...
3
votes
1answer
313 views

Hausdorff distance is a lower (or upper bound) for what probability metric?

In a metric space $X=(X, d)$, given a probability measure $\mu$ and two subsets $A$ and $B$ of positive measure, it's not hard to prove that $$ d(A, B) \le W(\mu|_A, \mu|_B), $$ where $d(A, B):= \...
0
votes
0answers
144 views

Is there a precise relationship between ``Geometric Functional Analysis" and high-dimensional probability/information theory?

The 2009 course on GFA by Roman Vershynin (https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf) introduced the subject with this line on the course page, "...
5
votes
1answer
163 views

General distributions with the “transportation-cost inequality” property to piece log-concave distributions

It is now known [Otto et Villani 2000; Cordero et al 2006; etc.] that on an $n$-dimensional smooth Riemannian manifold $X$ and a probability measure $\mu$ on $X$ with density $d\mu \propto e^{-V}dvol$ ...
4
votes
1answer
530 views

What is the nearest-neighbor distribution in this picture?

Consider the following process: sample $n$ points uniformly at random in the unit square, and for each point $i$, let $d_i$ be the distance from $i$ to its nearest neighbor. Finally, let $z_i = d_i\...
4
votes
1answer
83 views

Distribution of the $pn$ shortest edges out of $n$ uniform points, $p\to 0$

Suppose I sample $n$ points independently and uniformly at random in the unit square, and then I select the $pn$ shortest edges between all pairs of points, for fixed $0<p<1$. For large $n$ and ...
1
vote
0answers
99 views

Upper bound for $\sup_{W_d(Q, P) \le \epsilon} Q(A)$, where $W_d$ is the Wasserstein metric

Let $X=(X,d)$ be a metric space and let $W_d$ denote the Wasserstein metric induced by this metric, on the space of probability distributions on $X$. Let $\epsilon \ge 0$, $A$ be a Borel subset of $X$...
2
votes
0answers
56 views

Extreme points in $d$-dimensional quadrant hull of $n$ random points in a halfplane

Finding the asymptotic growth of the expected number of extreme points of the quadrant hull in $d$ dimensions seems to be a well studied problem. Interestingly, the solution appears to vary widely by ...
6
votes
2answers
563 views

Approximation of Wasserstein distance between $p_\theta$ and $p_{\theta + d\theta}$

Given a parametric family of distributions $\{p_\theta\mid\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid $$\operatorname{KL}(p_\theta\...
2
votes
2answers
107 views

Does this formula for caliper diameter hold for concave polyhedra?

I recently asked on MathOverflow and also asked several people I know to prove the following: How do I prove that the average caliper diameter of the polyhedron across all possible rotations is ...
26
votes
5answers
1k views

Moments of area of random triangle inscribed in a circle

The $2m$th moment of the (random) area of the triangle whose vertices are three independent, uniformly distributed random points on the unit circle appears to be $((3m)!/(m!)^3)/16^m$. Can anyone ...