All Questions
Tagged with pr.probability geometric-probability
164 questions
4
votes
0
answers
52
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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 ...
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
13
votes
1
answer
484
views
A probability involving areas in a random pentagram inscribed in a circle: Is it really just $\frac12$?
This question was posted at MSE but was not answered.
The vertices of a pentagram are five uniformly random points on a circle. The areas of three consecutive triangular "petals" are $a,b,c$...
- 3,527
13
votes
1
answer
761
views
If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{\pi^2}\chi_2(x) + \frac{4}{\pi^2}\chi_2(y)$
Posting this question in MO since it is unanswered in MSE
Let $(a,b,c)$ be the side of a triangle. In its most general linear form, the triangle inequality can be expressed as: Does $ax + by \ge c$ ...
- 5,470
11
votes
1
answer
746
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Find the area of the region enclosed by $\sin^p x+\sin^p y=\sin^p(x+y)$, the $x$-axis and the $y$-axis (comes from a probability question)
Consider the graph of $\sin^p x+\sin^p y=\sin^p(x+y)$, where $x$ and $y$ are acute, and $p>1$.
Here are examples with, from left to right, $p=1.05,\space 1.25,\space 2,\space 4,\space 100$.
Find ...
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13
votes
8
answers
1k
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The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$
The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
There is a convoluted proof that $P(ab>c)=\frac12$. But since the ...
- 3,527
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
2
answers
195
views
Random partition of an interval – Dirichlet distributed?
Let $X_1, \ldots, X_N \sim \operatorname{Unif}[0,1]$ and consider the intervals between successive order statistics: $[0, X_{(1)}], [X_{(1)}, X_{(2)}], \ldots, [X_{(N)}, 1]$.
What is the distribution ...
- 57
14
votes
1
answer
1k
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A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?
A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points.
For example, here are $20$ random ...
- 3,527
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
2
votes
1
answer
275
views
Does this KL divergence inequality hold?
Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that
$$
\frac{y\left( x \right)}{p\left( x \right)}=\frac{\...
- 49
2
votes
0
answers
306
views
Why is it impossible to create a numerically balanced die with more than 120 sides?
I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...
2
votes
0
answers
185
views
An attempt to define expected value of a Riemannian manifold valued random variable - what'll go wrong?
Let $X:\Omega\to (M,g)$ be a random variable taking values in a Riemannian manifold $(M,g)$ with the Riemannian volume form denoted by $dvol_g(x).$ We know that there's no standard way to generalize ...
- 1,467
7
votes
1
answer
310
views
Local Lipschitzness of parameterization of Gaussians in Wasserstein space
Fix a positive integer $n$ and consider the $2$-Wasserstein space $\mathcal{P}_2(\mathbb{R}^n)$. Let $X$ be the cone of $n\times n$ symmetric positive semidefinite matrices with Frobenius norm and ...
- 169
0
votes
1
answer
119
views
How many samples do you need to get constant dispersion?
Let $C_n$ be the hypercube $[-1,1]^n$. For $a_1,\cdots,a_s \in C_n$, define its dispersion $D(a_1,\cdots,a_s)$ as $\max_{x \in C_n}\min_{i \in [s]} \|x-a_i\|_{2}$. Let $0< \lambda < 1$ be a ...
- 261
2
votes
1
answer
201
views
Dispersion of a "random" subset of $[-1,1]^2$
Let $C$ be the square $[-1,1]^2$. Let $a_1,\dots,a_m$ be points chosen independently and uniformly at random from $C$. Let $d_m$ (dispersion) be the random variable $\max_{x \in C}{\min_{j \in [m]}{\|...
- 261
0
votes
1
answer
91
views
Can a measure on a finite metric space be Alhfors regular?
Recall that a probability $\mu$ measure on a metric space $(X,d)$ is called Ahlfors $q$-regular if there are $0<c\le C$ such that: for $\mu$-a.e.\ $x\in X$ one has
$$
cr^q \le \mu(B(x,r)) \le Cr^q,
...
- 364
0
votes
1
answer
115
views
Order of orthant probabilities in a prolate multinormal distribution
This is inspired by the negative answer to the conjecture in Which orthant probabilities are the largest? (For a multivariate normal distribution).
Suppose $X$ has the $k$-dimensional multivariate ...
- 4,219
2
votes
1
answer
81
views
Distribution of scaled Johnson-Lindenstrauss transforms
Suppose that $\mathcal{D}$ is a Johnson-Lindenstrauss (JL) distribution on $\mathbb{R}^{r\times n}$ ($1 \le r \le n$), meaning that there exist constants $\epsilon, \delta \in(0,1)$ such that
$$
\...
- 269
1
vote
1
answer
229
views
Gaussian width of intersection of cube and ball in high-dimensional euclidean space
Let $d$ be a large positive integer and fix $r \ge 0$. Set $S := B_2^n \cap [-r,r]^d$, where $B_2^d$ is the euclidean unit-ball in $\mathbb R^d$. Finally, let $\omega(S)$ be the Gaussian width of $S$, ...
- 6,853
1
vote
1
answer
241
views
Integration by parts for indicator of a sphere to indicator of a ball
Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
2
votes
0
answers
131
views
Eigenvalues of Witten Laplacian induced by log-concave probability measure on manifold
Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\mu=e^{-V}d\mathrm{vol}_M$ be a log-concave probability measure on $M$, such that the pair $(M,\mu)$ verifies the so-called Bakry-Emery ...
- 6,853
0
votes
1
answer
247
views
Projecting a given point onto a random $2$-dimensional plane in more than $3$ dimensions
We are given $\mathbf{p}\in\mathbb{R}^d$, where $d\gg 1$. Let $\mathbf{v}$ be a point selected uniformly at random from the unit $(d-1)$-sphere $\mathcal{S}^{d-1}$ centered at the origin $\mathbf{0}\...
- 1,677
0
votes
0
answers
195
views
Upper-bound for bracketing number in terms of VC-dimension
Let $P$ be a probability distribution on a measurable space $\mathcal X$ (e.g;, some euclidean $\mathbb R^m$) and let $F$ be a class of funciton $f:\mathcal X \to \mathbb R$. Given, $f_1,f_2 \in F$, ...
- 6,853
1
vote
0
answers
223
views
Given p1=P(X1>X2), p2=P(X1=X2) and p3=P(X1<X2) (with necessarily p1+p2+p3=1), what is λ1 and λ2？ [closed]
In the following three equations : p1, p2, p3 are known, find λ1=? , λ2=?
(with necessarily p1+p2+p3=1)
$$ p_1 = \sum_{0 \leq k_2 \lt k_1 \in Z}(\frac {\lambda _1^{k_1}} {k_1!}\cdot{e^{-\lambda _1}})(\...
- 29
1
vote
1
answer
115
views
Approximating the probability of a half-space using random Voronoi diagrams
Fix a half-space $H = \{x_1 \geq 0: ~ (x_1,\dots,x_n) \in \mathbb{R}^n\}$. Let $p$ be a distribution with support in $\mathbb{R}^n$. I am interested in the following way of estimating the weight $p(H) ...
- 33
1
vote
1
answer
231
views
For $x_1,...,x_n$ iid random on sphere of radius $\sqrt{d}$ in $R^d$, what is a good upper-bound on min distance of $x_{n}$ from the other $x_i$'s?
Let $n$ and $d$ be large positive integers with $n \le d^\gamma$, for some absolute constant $\gamma>0$; i.e., $n$ is at most polynomial in $d$. Let $x_1,\ldots,x_n,x_{n+1}$ be drawn iid from the ...
- 6,853
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
$$
\alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(...
- 6,853
4
votes
1
answer
1k
views
Defining a probability distribution on each tangent space of a manifold?
I've been reading about probability on manifolds. What bothers me is that there's no clear way to generalize the Gaussian distribution to manifolds. If we instead assign a probability distribution to ...
- 518
2
votes
1
answer
149
views
Tightly placed sphere caps
Problem. Fix a positive real $r>0$ and a positive integer $n>0$. Consider an independent, identical sample $X_1, \ldots X_n$ drawn from the uniform distribution over the unit $d$-sphere $\mathbb ...
- 903
0
votes
2
answers
187
views
Time-derivative of integral over sub-level set $s(t) := \int_{f^{-1}((-\infty,t])}p(x)dx$
Let $\mu$ be a probability distribution on $\mathbb R^d$ with "sufficiently regular" density $p$. Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently regular" function. Finally, ...
- 6,853
2
votes
1
answer
297
views
Examples of "almost" Ahlfors regular measures
Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$ such that there are $c,C,d,D>0$ satisfying: for every $x \in \mathbb{R}^n$ and every $r>0$
$$
c r^d \leq \mu(B(x,r)) \leq Cr^D.
$$
Let'...
- 5,405
4
votes
1
answer
209
views
Is $\int_{-c}^c |A \cap (x + A)|\, dx$ maximized when the measurable subset $A \subseteq \mathbb R$ is an interval centered at the origin?
Let $A$ be a nonempty measurable subset of $\mathbb R$, with Lebesgue measure $|A|=1$, and let $c>0$. Define the scalar $I(A)$ by
$$
I(A) := \int_{-c}^c |A \cap (x + A)|\, dx,
$$
where $x+A := \{x +...
- 6,853
2
votes
1
answer
165
views
Existence of preferred direction for a random vector with arbitrary distribution on sphere, under a condition on its covariance matrix
Let $X$ be random vector on the unit-sphere $S_{n-1}$ in $\mathbb R^n$. We don't assume that the distribution of $X$ is uniform on $S_{n-1}$
I'm interested in proving the existence of a (...
- 6,853
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
2
answers
283
views
Building the Wasserstein space by pushforwards
Let $\mathbb{R}^d$ denote the $d$-dimensional Euclidean space, $\mathcal{W}_2(\mathbb{R}^d)$ denote the $2$-Wasserstein space with respect to the $d$-dimensional Euclidean space $\mathbb{R}^d$. Let $...
- 5,405
6
votes
3
answers
447
views
Isoperimetric inequality for $\epsilon$-expansion of a set only along a certain subspace
Let $\gamma_n$ be the standard gaussian distribution on $\mathbb R^n$. Let $V$ be a $k$-dimensional subspace of $\mathbb R^n$. Finally let $A$ be any (nonempty) Borel subset of $A$ with $\gamma_n(A) = ...
- 6,853
3
votes
1
answer
403
views
Regularity of transport map
Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ with first moment and suppose that both $\mu$ and $\nu$ have a densities with respect to the $n$-dimensional Lebesgue measure. Fix some ...
- 5,405
0
votes
2
answers
204
views
What is the limiting marginal distribution of a fixed number of coordinates of a random point drawn uniformly on large-dimensional sphere?
Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges ...
- 6,853
2
votes
2
answers
322
views
Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere
This question is related to: https://math.stackexchange.com/q/4270522/168758
Let $H_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the ...
- 6,853
4
votes
2
answers
343
views
Concentration of $k$-th pairwise distance of random points in a unit square
For $1\leq i \leq n$, let $X_i\sim \text{Uniform}(0,1)$, $Y_i \sim \text{Uniform}(0,1)$ be $n$ points chosen uniformly in the unit square. Denote the $k$-th smallest pairwise distances across the $n$ ...
- 662
0
votes
2
answers
534
views
Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$
Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$.
Question.
What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution ...
- 6,853
6
votes
1
answer
374
views
Almost evenly distributed spherical random vectors
Consider $n$ i.i.d spherically distributed random vectors $z_1 ,\cdots , z_n \sim \text{Unif}(\mathbb{S}^{d-1})$. What is the best lower bound on $n$ for which whp there exists a constant $c>0$ ...
- 315
6
votes
2
answers
497
views
Average distance of the mean of $n$ random complex numbers in a unit disc
Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers distributed uniformly and randomly over the unit disc $x^2+y^2 \leq 1$. Let $z$ be the complex number defined by the mean of the of these numbers,that ...
- 826
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
5
votes
2
answers
290
views
Bounding Brownian motion and an Ito process simultaneously
Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...
- 245
4
votes
2
answers
175
views
Almost independence of $x^\top a$ and $x^\top b$ for $x$ uniform on the sphere in $\mathbb R^d$ and $a,b \in \mathbb R^d$ with $a^\top b = 0$
Let $d$ be a large positive integer. Let $x$ be uniformly distributed on the unit-sphere in $\mathbb R^d$ and let $a$ and $b$ be perpendicular vectors in $\mathbb R^d$, i.e such that $a^\top b=0$. Let ...
- 6,853
3
votes
1
answer
206
views
Random planes separating points in $\mathbb{R}^3$
We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$...
- 1,677
5
votes
1
answer
191
views
Probability of gaps between coordinates of a random point on the sphere
Let $X=(X_1,\ldots,X_n)$ be a point chosen uniformly at random from the sphere $S^{n-1}\subseteq \mathbb R^n$. Given $a>0$, what is the probability that $|X_1|^2-|X_i|^2\geq a$ for all $i>1$? Is ...
- 741
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