All Questions
Tagged with pr.probability geometric-probability
40 questions with no upvoted or accepted answers
15
votes
0
answers
398
views
Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ with probability $1$?
On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random ...
- 3,527
7
votes
0
answers
152
views
Probability of landing inside the convex hull of previously sampled points
Let $\{X_i\}_{0\leq i\leq\infty}$ be i.i.d. random vectors in $\mathbb{R^d}$.
I would like to show that the probability of one point being in the convex hull of the others goes to one with the number ...
- 233
4
votes
0
answers
52
views
Isomorphism of Wasserstein space implies isomorphism of base spaces?
Assume $(X_i,d_i)$ are polish spaces (or compact metric spaces) for $i=1,2$.
Further assume that the 1- Wasserstein spaces $(P_1(X_1),W_1)$ and $(P_1(X_2),W_1)$ are isometrically isomorphic. Does that ...
4
votes
0
answers
259
views
Malliavin calculus and geometric interpretation of $\nabla \cdot ({\nabla F(x)}{\|\nabla F(x)\|^{-2}})$, with regards to the surface $S = \{F = 0\}$
Let $F:\mathbb R^n \to \mathbb R$ be a "sufficiently regular" function. For any $k \ge 1$ and $x \in \mathbb R^n$, define
$$
\alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(...
- 6,853
4
votes
0
answers
180
views
Volume difference in random approximation of polytope
The following easily stated problem has arisen in my research. However, it's outside my field, and I'm unfamiliar with the literature. I would greatly appreciate any references.
Let $K\subset \...
- 211
4
votes
0
answers
481
views
The probability distribution for the number of pairwise distances $\leq$ some threshold for points uniformly placed in a sphere
If I place place $N$ particles in a sphere of radius $R$, selecting positions across the sphere's volume with uniform probability, what is the exact probability distribution for the number of pairwise ...
- 41
3
votes
0
answers
229
views
Expectation of angle between two vectors in the image of a gaussian random matrix
Let $m$ and $n$ be large positive integers (going to infinity), and let $W$ be a random matrix of size $n \times m$ with iid entries from $N(0,1/m)$. Let $x,y \in \mathbb R^m$ be deterministic vectors,...
- 6,853
3
votes
0
answers
132
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 ...
- 137
3
votes
0
answers
253
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 ...
- 1,095
3
votes
0
answers
234
views
Are random convex polygons on a sphere themselves sphere-like?
Say $\mathbb{R}^n$ is divided by $k>n$ randomly chosen hyperplanes. Each connected region away from the hyperplanes is the intersection of $k$ half-spaces, so it is a convex cone. It is known that ...
3
votes
0
answers
85
views
Expected Largest Area with Random Lines
We take the unit square and have it divided by $n$ lines which are chosen randomly. We choose the lines as follows, choose one of the four sides of the square at random and then choose a random point ...
- 133
3
votes
0
answers
133
views
Partitioning a sphere with a random tessellation
I have a $d$-dimensional sphere of radius $1$. I now tessellate the space with $d$-dimensional cubes of side $L$ and uniform distribution of the origin of the tessellation. Thus the size of the ...
- 162
3
votes
0
answers
134
views
Algorithm to calculate moments of uniform distribution on convex polyhedra
There is system of linear inequalities
$$
Ax \leq K,
$$
$$
x\geq a, x\leq b.
$$
$A$ is $(n\times m)$-matrix, where $n\approx 100$ and $m\approx 10000$, $rank(A)=n$.
Suppose that on set of solutions ...
2
votes
0
answers
64
views
Limiting distribution of separated points in a unit square
Let $n$ and $r$ be fixed, and consider the following process, with $S=\emptyset$ to start:
For $i\in\{1,\dots,n\}$:
Sample a random point $X$ in the unit square.
If $X$ is a distance at least $r$ ...
- 4,049
2
votes
0
answers
306
views
Why is it impossible to create a numerically balanced die with more than 120 sides?
I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...
2
votes
0
answers
185
views
An attempt to define expected value of a Riemannian manifold valued random variable - what'll go wrong?
Let $X:\Omega\to (M,g)$ be a random variable taking values in a Riemannian manifold $(M,g)$ with the Riemannian volume form denoted by $dvol_g(x).$ We know that there's no standard way to generalize ...
- 1,467
2
votes
0
answers
131
views
Eigenvalues of Witten Laplacian induced by log-concave probability measure on manifold
Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\mu=e^{-V}d\mathrm{vol}_M$ be a log-concave probability measure on $M$, such that the pair $(M,\mu)$ verifies the so-called Bakry-Emery ...
- 6,853
2
votes
0
answers
174
views
Random sets of points and hyperplanes in high dimensions
We are given a set $X$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \in\mathbb{R}^d$ selected uniformly at random from the unit origin-centered ball $\mathcal{B}^{d}$.
Consider the random ...
- 1,677
2
votes
0
answers
68
views
Approximate any point of the interval $[-1/2,1/2]$ by the sum of $n$ iid uniform random variables from $[-1,1]$
Let $x \in [-1/2,1/2]$ and $X_1,\ldots,X_n$ be drawn iid from the uniform distribution on $[-1,1]$.
Question. Given $\varepsilon \ge 0$ an integer $k \in [1,n]$, what is a good lower-bound on the ...
- 6,853
2
votes
0
answers
57
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 ...
2
votes
0
answers
899
views
norm of projection of a random vector on the sphere onto a linear subspace
Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{D-1}$ and let $\mathcal{V}$ be a linear subspace of dimension $m$. Then it is known that the euclidean norm of the ...
- 398
2
votes
0
answers
242
views
What transformations preserve the von Mises distribution?
The von Mises distribution is entirely defined on the circle with a density given by
$$f(x) = (2\,\pi\, I_0(\kappa))^{-1} \exp(\kappa \cos(x-\mu))\ ,$$
where $x$ is in an arbitrary real interval of ...
- 236
1
vote
0
answers
143
views
$\newcommand\v{\operatorname{vol}_d(C}$Compact subsets of $ℝ^d$ which maximize $\inf_{|v|\le1}\dfrac{\v\cap(𝜀v+C))}{\v)}$ for fixed $\v)$ and $𝜀>0$
Let $\operatorname{vol}_d$ be the volume measure on $\mathbb R^d$ and let $B_d$ be the unit-ball. For $\varepsilon \ge 0$ and a compact subset $C$ of $\mathbb R^d$ with $\operatorname{vol}_d(C)>0$, ...
- 6,853
1
vote
0
answers
62
views
Existence of a bigeodesic in last passage percolation is $0$-$1$ event
On the bottom of page two of This paper, the authors remark the following:
'...by translation invariance and ergodicity, we know that existence of a bigeodesic is a $0−1$ event and hence it follows ...
- 371
1
vote
0
answers
100
views
Cartesian product of Poisson processes
Consider $n$ smooth, compactly supported functions $\phi_1,\dots, \phi_n \in C_c^\infty(\mathbf{R})$, and generate $n$ independent Poisson spatial processes $N_1,\dots,N_n$ on $\mathbf{R}$, each with ...
- 455
1
vote
0
answers
83
views
Given a large random matrix, how to prove that every large submatrix whose range contains a large ball?
Context. Studing a problem in machine-learning, I'm led to consider the following problem in RMT...
Definition. Given positive integers $m$ and $n$ and positive real numbers $c_1$ and $c_2$, let's ...
- 6,853
1
vote
0
answers
64
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 $k$...
- 6,853
1
vote
0
answers
334
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)^...
- 6,853
1
vote
0
answers
94
views
Measure of the boundary of the support of a certain function defined by an expectation
Suppose:
$\mathcal{S} = \{ S \in \mathbb{R}^d \ | \ S_i > 0, \forall i = 1,...,d \} $
$R$ is a random vector (on some probability space, $\Omega$) such that, $R: \Omega \to \mathcal{S}$.
$h : ...
- 111
1
vote
0
answers
234
views
Strong Dependence
I don't know if this definition has been already given.
Suppose $X$ and $Y$ are two random variables over finite alphabets $\mathcal{X}$ and $\mathcal{Y}$. We say $Y$ is strongly dependent on $X$ if ...
- 1,109
1
vote
0
answers
245
views
Random walk conditioned on sum and last step
Can anyone help with the following (slightly weird) random walk question? I have a random walk starting at $X_0 = 1 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$. ...
- 747
1
vote
0
answers
501
views
Distribution of random vectors
Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).
A vector $u\in ...
- 143
0
votes
0
answers
24
views
Is there a log-concave distribution not spherical symmetric s.t $ \langle X, \theta \rangle$ is almost normal for all directions $\theta$?
Klartag's results indicate that for a log-concave isotropic random vector, with high probability over $\theta$, $\langle X, \theta \rangle$ is close to a normal distribution.
It is known that for the ...
- 11
0
votes
0
answers
195
views
Upper-bound for bracketing number in terms of VC-dimension
Let $P$ be a probability distribution on a measurable space $\mathcal X$ (e.g;, some euclidean $\mathbb R^m$) and let $F$ be a class of funciton $f:\mathcal X \to \mathbb R$. Given, $f_1,f_2 \in F$, ...
- 6,853
0
votes
0
answers
330
views
Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere
Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
- 6,853
0
votes
0
answers
113
views
How much a probability distribution is non-uniform in a convex subspace of $\mathbb{R}^d$?
I know a number of (standard and well known) ways to measure the distance between two probability distributions and, more in general, to quantify how much one is far from another.
Could you please ...
- 1,677
0
votes
0
answers
321
views
Projecting a vector onto a random subspace
Let $A\in\mathbb{R}^{k\times d}$ be matrix with i.i.d. $\mathcal{N}(0,1/k)$ entries with $k<d$, and let $B=A^{\top}A$. I would like to compute the distribution of $Bx$ where $x\in\mathbb{R}^{d}$ is ...
- 129
0
votes
0
answers
165
views
Probability that the perturbed convex hull is larger than the original one
I am wondering if any convex geometers/probabilists have looked at the following question:
Given $n$ randomly distributed (not sure what assumption to put there) points in $\mathbb{R}^d$, for each ...
- 133
0
votes
0
answers
137
views
Expected Number of Triangles
A unit square is divided up with $n$ random lines. The random lines are chosen as follows, we choose one side of the square and pick a random point on that side. From there we choose a random point on ...
0
votes
0
answers
454
views
Reference: Bochner Integral`
What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
- 5,405