All Questions
Tagged with geometric-measure-theory pr.probability
46 questions
10
votes
1
answer
1k
views
Extension of measures from the ball sigma-algebra to the borel sigma-algebra
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
- 307
9
votes
2
answers
706
views
Measures whose projections are absolutely continuous
Since my question was not answered on MSE, I would like to ask it here.
Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt ...
- 452
9
votes
2
answers
2k
views
common dominating measure for a family of measures
Given a family $\{\mu \}_{i\in I}$ on a Polish space (complete, separable metric space) $X$. When does there exist a measure $\lambda$ such that
$$\mu_i=f_i \lambda$$
where the $f_i$ are densities (...
- 1,256
8
votes
2
answers
849
views
Is the Gaussian Correlation Inequality universal?
T. Royen proved the Gaussian correlation inequality in the context of Gamma distributions back in 2014, which was since popularized by Latala and Matlak. The properties of Gaussian integration seem ...
- 4,466
7
votes
1
answer
700
views
About taking an expectation over orthogonal matrices
Say $Q$ is a random variable which is sampling orthogonal matrices in $m$ dimensions using the Haar measure on $O(m)$. Let $A$ and $B$ be some (fixed) subset of rows and columns of $Q$ such that $\...
- 2,246
7
votes
1
answer
424
views
Transportation-cost inequality for pushforward measure
Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies ...
- 6,853
6
votes
1
answer
730
views
Supremum of a stochastic process
Let $(x_1,...,x_N)$ be points in $R^d$, and $\sigma=(\sigma_1,...,\sigma_N)$ are i.i.d. Rademacher variables (+1 or -1 with probability 0.5 each). (Or alternatively, $\sigma$ could be a standard ...
- 517
5
votes
2
answers
950
views
Comparison of Information and Wasserstein Topologies
There are many possible metrics one can place on the space of Gaussian probability measures on $\mathbb{R}^n$, with strictly positive definite co-variance matrices. Let's denote this space by $X$.
I'...
5
votes
0
answers
143
views
Smoothing properties of convolutions of $P^1(\mathbb{R})$ by $SL(2,\mathbb{R})$
Consider the action of $SL_2(\mathbb R)$ on real projective space $P^1(\mathbb R)$; given $A \in SL_2(\mathbb R)$ and $\alpha \in P^1(\mathbb R)$ we write $A . \alpha \in P^1(\mathbb R)$ for this ...
- 485
4
votes
1
answer
487
views
Finiteness of Hausdorff measure of balls
Let $(X,d)$ be an arbitrary metric space and let $\Bbb B(x,r)$ denote the closed ball with center $x \in X$ and radius $r>0$. For $p\geq 0$, let $H^p$ denote the $p$- dimensional Hausdorff measure. ...
- 185
4
votes
2
answers
10k
views
Random packing density of cylinders in a volume
I am trying to calculate the packing density of cylindrical bottles in a box, assuming that the bottles are randomly dumped in the box.
I have read on the packing density of spheres here https://en....
- 143
4
votes
2
answers
258
views
What theorem can be used to explain this occurrence?
I'm not highly versed in research-level mathematics. I do conduct research in cellular biology. I was wondering if you could help me find a term that can be referred to when discussing the following ...
- 49
4
votes
1
answer
265
views
Bounds on discrepancy metric of product measures
Consider two measurable spaces $X_1 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_1)$ and $X_2 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_2)$ and the product spaces
$$X_1^{q} = (\times_{i=1}^q\...
- 2,712
4
votes
0
answers
66
views
Tangent distribution for particular non-doubling measure (GMC)
The radon measure $\mu$ on [0,1] called GMC (Gaussian multiplicative chaos) satisfies the following:
$$(1/c)|I|^{a}\leq\mu(I)\leq c|I|^{b},$$
$$\sup_{x\in [0,1]}\frac{\mu(B_{2r}(x))}{\mu(B_{r}(x))^{1-...
- 5,474
3
votes
2
answers
293
views
Wasserstein convergence of "series expansion'' of probability measure
Let $X$ be a Polish space and let $(\mu_i)_{i=1}^{\infty}$ be a sequence of probability measures in the Wasserstein space $\mathcal{P}(X)$ on $X$. Let $(\beta_i)_{i=1}^{\infty}$ be a summable ...
- 5,405
3
votes
1
answer
190
views
Example where concentration of measure fails nontrivially
A metric probability space $(X, \mu, \rho)$, i.e., a complete separable metric space with a probability measure on its Borel sets, is said to satisfy (Gaussian) concentration of measure property if ...
- 141
3
votes
3
answers
700
views
Uniform distribution with respect to Hausdorff measure
Suppose I have some nicely defined "fractal" subset of (to make life simpler) Euclidean space $\mathbb{E}^n,$ of some arbitrary Hausdorff dimension $s,$ such that the corresponding Hausdorff measure $...
- 96.4k
3
votes
1
answer
77
views
Continuous selection parameterizing discrete measures
Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of ...
- 5,405
3
votes
1
answer
188
views
Equivalent Definitions of the Gaussian Surface Measure for Regular Sets
I wonder if the following definitions of the Gaussian surface measure are equivalent.
First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e....
- 1,127
3
votes
0
answers
185
views
Measure change bound for function of subgaussian r.v
Let $X$ be a (sub)gaussian r.v. on $\mathbb{R}^d$; say $X\sim\mathcal{N}(\mathbf{0},\mathbb{1}_d)$; and let $a\colon\mathbb{R}^d\to [0,1]$ be a function with $\mathbb{E}[a(X)] > 0$.
It is not hard ...
- 1,372
2
votes
1
answer
1k
views
Wasserstein interpolation between two probability measures on a metric space
Question 1
Given probability measures $\mu$ and $\nu$ on the same metric space $X=(X,d)$, and $\alpha \in [0, 1]$, is it always possible to find another probability measure $\lambda_\alpha$ on $X$ ...
- 6,853
2
votes
1
answer
241
views
Weak continuity of law
Let $\mathcal{P}_2(\mathbb{R}^n)$ denote the set of all Borel probability measures on $\mathbb{R}^n$ with finite variance and weak topology. Let $X_t$ be a strong solution to the SDE with initial ...
- 5,405
2
votes
1
answer
155
views
Covering of discrete probability measures
Let $\mathcal{P}_{n:+}(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$ where $k_i>0$. Then any measure in $\mathcal{P}_{n:+}(\...
- 5,405
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
2
votes
0
answers
29
views
Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
- 434
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
79
views
Compute Mixed Volume with Respect to Some Regular Sets
Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
- 1,127
1
vote
1
answer
121
views
Relaxation of requirements for Anderson's inequality
Anderson's inequality states that for a nonnegative, symmetric, globally integrable and unimodal function $f$, i.e.
$f(x) \geq 0$,
$f(-x) = f(x)$,
$\int f(x) dx < \infty$
For all $t\in \mathbb R$, ...
- 375
1
vote
1
answer
176
views
Is an inner product $\langle X, \epsilon\rangle$ between log-concave $X$ and $\epsilon\gets \{0,1\}^n$ log concave?
Let $X$ be a random variable with a density $p(x)$ with respect to the Lebesgue measure. We say that $X$ is log concave if $p(x) = \exp(-V(x))dx$ for $V(x)$ a convex function.
Let $X$ be log-concave ...
- 2,183
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
172
views
A question about pushforward measures and Peano spaces
Specifically my question is the following: Let $P$ be a Peano space. If $(P,\sigma,\mu)$ and $(P,\sigma,\nu)$ are both nonatomic probability measures, does there exist a continuous function $f:P\to P$ ...
- 33
1
vote
1
answer
88
views
Convergence of probability measures which (asymptotically) concentrate along a submanifold
Let $V : (-1, 1)^d \to \mathbf{R}_+$ be a smooth function, and for $\beta > 0$, define
\begin{align}
P_\beta ( dx ) &= \exp \left( - \beta V ( x ) \right) / z (\beta) \, dx\\
z (\beta) &= \...
- 801
1
vote
0
answers
87
views
Symmetry of the isoperimetric profile
Given a probability measure $\mu$ on a metric space $(X, \mathsf{d})$, the $(\mu-)$Minkowski content of a set $A$ is defined as
$$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$...
- 801
1
vote
0
answers
199
views
Absolute continuity of joint distribution if all marginals in any basis are absolutely continuous
Consider a probability distribution $\nu$ on $(x,y)\in\mathbb{R}^2$. I know that the absolute continuity of the marginals on $x$ and $y$ is not sufficient to imply the absolute continuity of $\nu$, ...
- 449
1
vote
0
answers
105
views
Gaussian isoperimetry for $\ell_p$ norms
Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$. It is well-known (e.g see Proposition 1) that for a given Gaussian volume content, half-spaces $H=\{x \in \mathbb R^n | a^Tx \le b\}$ ...
- 6,853
1
vote
0
answers
67
views
Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$
Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
- 19
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
96
views
Random projection increases the distance?
Consider two absolutely continuous random variables $X: \Omega \mapsto \mathbb{R}^d$ and $Y: \Omega \mapsto \mathbb{R}^d$ for probability spaces $(\Omega, \mathcal{F},p_X)$ and $(\Omega, \mathcal{F},...
- 482
1
vote
1
answer
150
views
Probability Content of a random ball in R^n
As a follow up to this question, concerning this paper:
Given random variables $X_1,\ldots,X_N,X_q:\Omega\rightarrow\mathbb{R}^d$, where $X_1,\ldots,X_N$ are independent and identical distributed. ...
- 329
0
votes
1
answer
132
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
$$
\int_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \...
- 5,405
0
votes
1
answer
189
views
Visualization of the disintegration theorem [closed]
Where can I find a picture that gives a visualization of the disintegration theorem?
If such reference does not exist, what would a nice visualization of this fundamental result look like?
- 109
0
votes
0
answers
77
views
Wasserstein space isomorphic to original space?
Is there a complete measurable metric space $(X,d)$ for which its $p$-Wasserstein space $W(X)$ is isometrically isomorphic to $(X,d)$ for some $p \in [1,\infty]$?
Note that there is a canonical non-...
0
votes
0
answers
83
views
Distortion estimates to control Hausdorff measure of a curve
I am studying the paper Blumenthal - Statistical properties for compositions of standard maps with increasing coefficent.
I have a problem to understand how the distortion estimates are used. The ...
0
votes
0
answers
59
views
Examples of strongly continuous measure-valued functions
Let $X$ be a compact geodesic metric space and let $P_p(X)$ be the set of all finite Borel measure on X with finite $p^{th}$ moment. We equip $P_p(X)$ with the total variation topology metric. What ...
- 5,405
0
votes
0
answers
96
views
If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?
Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
- 167
0
votes
0
answers
57
views
Absolute continuity of probability measures determined by dependence structure
We are on $\mathbb{R}^d$ with Borel $\sigma$-algebra. Let $\mu_1, ..., \mu_d$ be probability measures on $\mathbb{R}$ and $\Pi(\mu_1, \mu_2, ..., \mu_d)$ be the set of probability measures on $\mathbb{...
- 1,095