# Questions tagged [geometric-probability]

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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....