# Questions tagged [geometric-probability]

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156
questions

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17 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 ...

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125 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 :...

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122 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 ...

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67 views

### Local variation of normals on Riemannian submanifolds of $\mathbb R^D$

Let $M^d\subseteq\mathbb R^D$ be a compact manifold, $p\in M$ and $\nu\in N_p(M)$. Suppose that all the principal curvatures in direction $\nu$ are in $[a,b]$ for some $0<a<b<\infty$. Suppose ...

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46 views

### Lebesgue measure of a circular arc on the surface of the hypersphere [closed]

The following sub-problem came up in one of my research works:
Suppose that $U$ is a uniform random variable on the surface of the $d$-dimensional sphere with center at the origin and of radius $r$. ...

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**1**answer

147 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 $H$...

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164 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 ...

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96 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 ...

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122 views

### Is disintegration continuous?

Let $X,Y$ be Polish spaces and suppose that $X$ is compact. Denote by $\mathcal{Mes}(X,\mathcal{P}(X\times Y))$ the set of (Borel) measurable functions from $X$ to the set of Borel probability ...

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863 views

### Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments

It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...

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106 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 ...

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**1**answer

76 views

### Geometric sampling problem in the Euclidean space in high dimensions

Let $T$ be the triangle whose vertices are three given points $\mathbf{x}, \mathbf{y}, \mathbf{z}\in\mathbb{R}^d$.
Question: What computationally efficient strategy can we use to sample a point $\...

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**1**answer

152 views

### Does there exist a scale invariant random packing of circles in the plane?

I want to construct a scale invariant random packing of the plane with circles.
Here is a way to construct a rotationally invariant, but not scale invariant random packing of the plane with circles:
...

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**1**answer

168 views

### Iterating projections to random halfspaces

Consider the following process:
Start with a set $S = \mathbb R^n$. Repeat $L$ times: choose a random orthonormal basis $u_1, \ldots, u_n$, and consider the cone $C = \{ \sum \alpha_i u_i : \alpha_i \...

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157 views

### Sampling method for a specific distribution in high dimensions

We are given a set $X$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\in\mathbb{R}^d$, where $d\ll n$. Given any point $\mathbf{p}$ on the unit $(d-1)$-sphere $\mathcal{S}$, we define
...

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86 views

### Parameterization of exponential family

Let $\{\mathbb{P}_{\theta}\}_{\theta}$ be an exponential family of probability measures, all with finite mean. Under what conditions is the parameterization map $\theta\mapsto \mathbb{P}_{\theta}$ ...

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101 views

### Barycenters on Hadamard Manifolds

Let $(M,g,m_0)$ be a pointed-Hadamard manifold with Riemmanian distance function $d_g$, $(X,\Sigma,\mu)$ be a finite measure space. We use $L^2(\mu;M,m_0)$ to denote the metric space consisting of ...

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153 views

### Expected triangle area of normal distributed vertices with colinear expectations

For the bounty the already answered problem was reformulated
This question was already answered for random variables in $\mathbb{R}^3$. Now I am looking for the solution in $\mathbb{R}^2$ that could ...

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79 views

### What kinds of gradient-flows on $\mathbb R^d$ preserve the log-concavity of the distribution $\mu_0$ of starting point $x_0$

Let $\mu_0$ be a log-concave distribution on $\mathbb R^d$ and let $f:\mathbb R^d \to \mathbb R$ be $C^2$. Let $x_0$ be sampled uniformly at random from a log-concave distribution $\mu_0$, meaning ...

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84 views

### Lower-bound for $\underset{p \le \gamma_d(A) \le q}{\inf} \gamma(A^\epsilon)$, where $\gamma_d$ is the standard gaussian distribution on $\mathbb R^d$

Let $\gamma_d = \gamma_1 \otimes \ldots \otimes \gamma_1$ be the standard Gaussian distribution on $\mathbb R^d$, where $d$ is a large positive integer. Given $\epsilon \ge 0$ and a measurable $A \...

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102 views

### Lower-bound on Sobolev norm of function on $(d-1)$-dimensional sphere, whose sign has been fixed at $n$ points

Let $\mathbb S_{d-1} := \{x \in \mathbb R^d \mid x^\top x = 1\}$ be $(d-1)$-dimensional sphere in $\mathbb R^d$ and let $\sigma_d$ be the uniform distribution on $\mathbb S_{d-1}$. Let $x_1,\ldots,x_n$...

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55 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 ...

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55 views

### Distribution of line segment intersections in random pointsets

let $P$ be a set of $n$ points that are uniformly distributet inside the unit square ore unit circle, and $L=\lbrace\ell_{ij}\rbrace := \lbrace \lbrace \alpha p+ (1-\alpha q)\rbrace\,|\,0\le\alpha\le ...

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108 views

### Edges in the convex hull of the union of random polygons

Let $P$ and $Q$ be two convex polygons in $\mathbb{R}^2$. Given $a > 0$, denote by $aP$ its image under the dilation by $a$ centered around the origin (i.e. the polygon obtained by replacing each ...

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58 views

### Projection onto manifold of Gaussian measures by “trunction” of moments

Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of Borel probability measures on $\mathbb{R}^n$ with finite mean and variance; in the sense that
$$
\int_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \...

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57 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 ...

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99 views

### Question on limit in probability of the ratio of max to min of 2 sequences of non-ive, continuous iid random variables with support $[0, \infty).$

For each $ m \ge 1$, let $X_m$ and $Y_m$ be two non-negative iid random variables with the same distribution. (The distributions of $X_m$ may change with different $m$.)
**Assume that their support of ...

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**1**answer

63 views

### Mass distributions for high dimensional simplex and cross polytope

In this question, it is shown that the radial mass distribution of an $n$-cube (i.e. the probability density for the distance from a point selected uniformly from within an $n$-cube to the cube's ...

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87 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,...

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48 views

### Characterization of random variables whose tensor powers have subexponential “small-ball” probabilities

Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties
1. Symmetry: $\zeta \overset{d}{=} - \zeta$.
2. Small-ball probability: there exists ...

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485 views

### Taking points uniformly inside a general finite geometric domain

It is well known that if we want to take $n$ uniformly and randomly points inside a circle of radius $r$ and centered at the origin the following apparently correct approach for generating $x$ and $...

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116 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 ...

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**1**answer

142 views

### How tight is the bound $P(\|X\|^2 \ge t |\langle a,X\rangle|) \ge 1 - t\sqrt{\frac{2}{m-1}}$, where $X \sim N(0, I_m)$ and $\|a\| = 1$?

Let $X$ be a random vector in $\mathbb R^m$ with iid $N(0,1)$ coordinates and let $a$ be a fixed unit vector in $\mathbb R^m$. In another post (SE link here https://math.stackexchange.com/a/3792730/...

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280 views

### What is the probability that a random chord in a sphere touches opposite hemispheres?

(edited) Consider the unit sphere $\mathbb{S}^2\subset \mathbb{R}^3$, and its upper $(z>0)$ and lower $(z<0)$ hemispheres.
Draw two independent, uniformly distributed points $X,Y$ on $\mathbb{S}^...

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80 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 ...

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85 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 ...

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66 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 ...

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2k views

### A variation of the law of large numbers for random points in a square

I uniformly mark $n^2$ points in $[0,1]^2$. Then I want to draw $cn$ vertical lines and $cn$ horizontal lines such that in each small rectangle there is at most one marked point. Surely, for a given ...

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89 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 ...

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98 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 ...

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579 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 $k < n$ ...

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63 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 $k$...

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**1**answer

187 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, \...

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66 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 ...

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**1**answer

82 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 $Q$.
...

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**1**answer

81 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 ...

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votes

**1**answer

129 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....

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**2**answers

497 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 ...

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109 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)^...

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**1**answer

105 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 \...