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2
votes
0answers
47 views

Intrinsic volumes of a family of convex sets $\{K_n\}$

Consider a family of convex sets $\{K_n\}$ such that $K_n \subset \mathbb{R}^n$ for each $n$. The kinds of sets one might be considering could be, for instance, $K_n$ is the cube of side $2A$, ...
5
votes
3answers
214 views

Probability that convex hull of multivariate Gaussian sample contains a given point

I am generating random vectors $X_1, \dots, X_N$ from a $d$-dimensional multivariate normal $\text N(\mu, \Sigma)$. I would like to know what is the probability that a given point $y \in R^d$ falls ...
4
votes
2answers
564 views

Do Random Walks on the Hexagonal Lattice have a limit?

For every positive integer $n$, consider a regular hexagon $\mathrm{H}_n$ such that the distance of each vertex from the center is $\frac{1}{\sqrt{n}}$. That in turn induces a tiling of ...
8
votes
1answer
277 views

Can one use Brownian motion to prove that two manifolds are not conformally equivalent?

Let me start by a very simple example; consider the following question: "Let D1 be a square and D2 a rectangle (boundary included). View them as subsets of the complex plane. Does there exist a ...
0
votes
0answers
92 views

Area on the unit sphere swept out by big circles corresponding to a curve

For a point on the unit sphere, we know the plane perpendicular to the line through the origin and the point cuts the sphere with a big circle. When the point moves along a sphere curve, the ...
11
votes
1answer
232 views

Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube.

Consider an integer cube of size $\sqrt{k} \times \sqrt{k} \times \sqrt{k}$, where $k$ is an asymptotically large perfect square number. Place k points in this cube at uniformly random locations, ...
1
vote
1answer
130 views

Distance between poisson points in two disjoint unit discs

Two disjoint unit discs $D_1$ and $D_2$. Inside them there are random poisson points with intensity $\lambda$. For a given real $r>0$, what is the probability that there exist a poisson point ...
4
votes
1answer
119 views

Nontrivial lower bounds on Cheeger inequalities for Markov chains

For a reversible Markov chain $X_{t}$ on $\mathbb{R}^{n}$ with transition kernel $K$ and stationary distribution $\pi$, it is well-known that the `spectral gap' (basically, the size of $K$ when ...
1
vote
0answers
128 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 $[-1,1]$. ...
1
vote
1answer
82 views

Equivalence between choosing a subspace and choosing its orthogonal

Hi, We consider subspaces of $\mathbb{R}^N$. Suppose that we have a property called $\mbox{Prop}$ that apply to subspaces of $\mathbb{R}^N$. That is to say a function from the set of subspaces of ...
1
vote
0answers
234 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 ...
1
vote
1answer
292 views

Product of densities of a wrapped normal distribution

The density of a wrapped normal distribution is given by $$\frac{1}{\sigma \sqrt{2\pi} }\sum _{k=-\infty }^{\infty }\text{Exp}\left[\frac{-(\theta -\mu -2\pi k)^2}{2\sigma^2}\right]$$ Considering two ...
7
votes
2answers
436 views

Fitting a mesh to a density function

Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). For large $N$, is ...
3
votes
1answer
337 views

Cover a line segment randomly with smaller line segments

Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon). But the problem when the circle is changed to a line segment doesn't seem to have been ...
3
votes
1answer
203 views

Distribution of shapes of Delaunay triangles

People haven't been rushing in to answer this question I asked on stackexchange yesterday. The way I phrased it initially was this: Does anyone know the probability distribution of the shapes of ...