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10 votes
1 answer
337 views

What are the predictive implications of conditional non-commutative probability?

To simplify things, let's consider the Hilbert approach to quantum probability over a finite dimensional vector space $V$ of dimension $n$. In this context a state $S$ is a positive semi-definite ...
Mehmet Coen's user avatar
10 votes
1 answer
330 views

(Sharp) Bounds on $E(XYZ)$ given all the bivariate marginals

Suppose $X,Y,Z$ are all real-valued random variables. Suppose I know the joint marginal distributions of $(X,Y)$, $(Y,Z)$ and $(X,Z)$. I want to find bounds on $E(XYZ)$. In the case of bounding $E(XY)$...
Atom Vayalinkal's user avatar
9 votes
3 answers
868 views

Rosenthal like inequality for weak $\mathbb L^p$-norms

Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if $(X_i)_{i=...
Davide Giraudo's user avatar
9 votes
1 answer
4k views

What are some characterizations of the strong and total variation convergence topologies on measures?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here. The Wikipedia article on convergence of measures defines three kinds of convergence: ...
user39080's user avatar
  • 203
9 votes
1 answer
652 views

Scaling in Mehta's integral

The following expression is known as Mehta's integral and deeply connected to random matrix theory: $$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
Pritam Bemis's user avatar
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 ...
limanac's user avatar
  • 452
9 votes
3 answers
2k views

2-Wasserstein (optimal transport) and extension to the set of all signed measures

Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as $$ d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2} $$ ...
passerby51's user avatar
  • 1,731
9 votes
1 answer
359 views

Relaxation of notion of positive definite function

A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
Hans's user avatar
  • 3,031
9 votes
2 answers
548 views

What mode of convergence is this?

I'm interested in a new (to me) mode of convergence which is stronger than convergence in measure/probability. I want to know if it has a name and if it is used much in the literature. I will write ...
Jason Rute's user avatar
  • 6,287
9 votes
1 answer
450 views

A question on infinite dimensional Gaussian measure and affine tranformations.

Let $\gamma_\infty$ denote the product Gaussian measure on $\mathbb{R}^\mathbb{N}$. Which $a,b \geq 0$ satisfy that for every Borel set $K\subseteq \mathbb{R}^\mathbb{N}$ of positive measure, $a K + ...
Justin Moore's user avatar
  • 3,547
9 votes
1 answer
385 views

A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as $$\rho_2(X;Y):=\sup\frac{\mathbb{E}[f(X)g(Y)]}{||f||_2||...
math-Student's user avatar
  • 1,109
9 votes
3 answers
654 views

measure with given push-forwards

Let $X,Y$ be locally compact spaces (in my specific case, they are locally compact groups). Suppose that we are given a measure $\mu$ on $X$ and a finite number of quotient maps $p_1,\ldots,p_n:Y\...
steven deprez's user avatar
9 votes
2 answers
674 views

Small crown probabilities (and infinite dimensional margin assumption)

My question is: How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two. Notations and definitions (to make the question rigorous) Let ...
9 votes
0 answers
305 views

Convergence in $L^2$ of iterated expectations

Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$. Define the iterated expectations of X as follows: $X_0 = X$, and, ...
Ben Golub's user avatar
  • 1,068
8 votes
2 answers
1k views

Talagrand's inequality for the discrete cube

Talagrand showed that if $f$ is a convex $1$-Lipschitz function on $\mathbb{R}^n$, and if $\mu$ is a product of probability measures supported over the interval, then $f$ has Gaussian concentration w....
alesia's user avatar
  • 2,772
8 votes
5 answers
685 views

Distributions of distance between two random points in Hilbert space

Let $\mu$ be a probability distribution on a separable infinite-dimensional Hilbert space. Let $D$ be the distance between two independent random samples from $\mu$. So $D$ has some probability ...
David Aldous's user avatar
8 votes
2 answers
644 views

Uniqueness of the uniform distribution on hypersphere

I'm looking for a uniqueness-type result for the following problem, which is related to the uniform distribution in the hypersphere $\mathbb{S}^{p-1}$. Suppose $f$ is a sufficiently smooth function on ...
pat2211's user avatar
  • 81
8 votes
2 answers
1k views

Does infinite-dimensional Brownian motion live in hyperplanes?

I'll begin this question with the finite-dimensional case, as a warmup. Let me say a continuous path $\omega : [0,1] \to \mathbb{R}^d$ is hyperplanar if there exists a nonzero $x \in \mathbb{R}^d$ ...
Nate Eldredge's user avatar
8 votes
1 answer
360 views

Can we recover a topological space from the collection of Borel probability measures living on it?

Let $(X, \tau)$ be a topological space, and $\mathcal{P}(X, \tau)$ be the Borel probability measures living on $X$. Can we recover $(X, \tau)$ from $\mathcal{P}(X, \tau)$?
user avatar
8 votes
2 answers
540 views

Maximum entropy priors in infinite dimensional spaces

Is there an extension of maximum entropy probability distributions for function spaces? For $\mathbb{R}^n$ and discrete spaces, there is much literature about this problem under names such as "non-...
Nick Alger's user avatar
  • 1,160
8 votes
2 answers
640 views

Does a random sequence of vectors span a Hilbert space?

Let $\mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $\mathcal{H}$ such that $P(v \perp h) < 1$ for all $h \in \mathcal{H}.$ Suppose we sample an infinite ...
J. E. Pascoe's user avatar
  • 1,429
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 ...
John Jiang's user avatar
  • 4,466
8 votes
1 answer
1k views

derivative in the Wasserstein space

Villani gives the following formula to find the gradient of a function $F$ of a probability density function $\rho$ in the Wasserstein space : $$\nabla_W F(\rho) = -\nabla.(\rho \nabla \frac{\delta F}{...
WhitAngl's user avatar
  • 481
8 votes
1 answer
1k views

Is there a regular Dirichlet form with no associated Feller process?

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda (hereafter, [FOT]). In Chapter 7, where they discuss the construction of a Markov process ...
Nate Eldredge's user avatar
8 votes
0 answers
182 views

Distribution domination for sums of independent random variables in Banach spaces

Let $X$ be a Banach space and let $(\xi_n)$ and $(\eta_n)$ be independent mean-zero random variables with values in $X$ satisfying $$ \sum_n \mathbb P(\xi_n \in A) \leq \sum_n \mathbb P(\eta_n \in A), ...
Iv Yar's user avatar
  • 131
8 votes
0 answers
211 views

Superharmonic functions and amenability

Let $G$ be a group generated by a finite set $S$. Let $P$ be a Markov operator defined by the uniform measure on $S$. A function is superharmonic if $Pf\leq f$. Assume that there is a set of non-...
Kate Juschenko's user avatar
7 votes
2 answers
988 views

Missing mass conjecture

Let $n,t$ be positive integers and $p_1,p_2,\ldots,p_n$ positive numbers summing to 1. Conjecture: $$ \sum_{i=1}^n p_i (1-p_i)^t \le \frac{n(1-1/n)^n}{t} $$ always holds. The motivation comes from my ...
Aryeh Kontorovich's user avatar
7 votes
4 answers
1k views

Is the space of tempered distribution second countable?

Let $\mathcal S '(\mathbb R^d)$ be the space of Schwartz tempered distributions equipped with the weak-* topology. I need to know if this space is second countable, i.e. if this topology has a ...
Thomas's user avatar
  • 630
7 votes
3 answers
4k views

Is a semicontinuous real function Borel measurable?

Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous function. [Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable? If not, can one find a counter-example? Note that, for any $c$, ...
kenneth's user avatar
  • 1,399
7 votes
2 answers
841 views

Why is $\mathbb R^{\mathbb N}$ not high-dimensional enough?

In this paper [1], the authors consider the limiting distribution of $$S_{n,p}:=\frac{1}{\sqrt n}\sum_{k=1}^nX_k$$ for $p\rightarrow\infty$ as $n\rightarrow\infty$, where $X_1, X_2,\dots, X_n$ are ...
Quertiopler's user avatar
7 votes
1 answer
1k views

If $H$ is a separable Hilbert space, is $L^2(H)$ separable?

Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$. Is the Hilbert space $L^2(H,\gamma)$ separable?
Tom LaGatta's user avatar
  • 8,512
7 votes
4 answers
946 views

On operator ranges in Hilbert & Banach spaces

Lemma 1 from Anderson & Trapp's Shorted Operators, II isLet $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent: (1) ran($A$) $\subset$ ...
Tom LaGatta's user avatar
  • 8,512
7 votes
1 answer
423 views

Best constant in comparison between Rademacher and gaussian averages?

Let $(g_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon_k)$ be a sequence of independent rademacher variables. What is the best ...
BigBill's user avatar
  • 1,222
7 votes
1 answer
1k views

Properties of convolutions

Consider the function $$f_{n}(x)=e^{-x^2}x^n.$$ and the function $$h_p(x):=e^{-\vert x \vert^p}.$$ My goal is to analyze $$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*...
Landauer's user avatar
  • 173
7 votes
1 answer
261 views

Comparison of several topologies for probability measures

Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
Kass's user avatar
  • 243
7 votes
2 answers
460 views

Gaussian Surface Area of Positive Semidefinite Cone

Let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e.g., one that has smooth boundary or is convex. We define the $\epsilon$-neighbor of $A$ in the ...
Minkov's user avatar
  • 1,127
7 votes
1 answer
319 views

Why aren't operator semigroups studied from a dynamical perspective?

Often times one talks about iterating a continuous map to get discrete topological dynamics, or having a 1-parameter family of continuous maps to get continuous topological dynamics. When studying ...
Jeff's user avatar
  • 277
7 votes
1 answer
719 views

Tightness and Functional Analysis

Let $(\Omega , \mathbb{P})$ be a probability space and $X$ be a real-valued random variable. Then we immediately have the push-forward measure $\mu$ on $\mathbb{R}$ and one can think of $\mu$ as an ...
George Shakan's user avatar
7 votes
1 answer
253 views

Are all quasi-regular points on Polish spaces generic points?

Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
Dominik Kwietniak's user avatar
7 votes
1 answer
1k views

Reference request: norm topology vs. probabilist's weak topology on measures

Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...
JohnA's user avatar
  • 710
7 votes
1 answer
444 views

Is an infinite-dimensional "Lebesgue measure" uniquely determined by a set of positive finite measure?

Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex. We say that $\mu$ admits shifts if ...
Alexander Shamov's user avatar
7 votes
1 answer
624 views

Expectation involving maximum of Gaussian variables

Let $X\sim N(0, I_d)$ be a $d$-dimensional Gaussian random vector. Let $W_1, \ldots, W_k \in \mathbb{R}^d$ be $k$ fixed vectors in general positions. It is clear that $w_i^\top X, \ldots, w_k^\top X$ ...
Steve's user avatar
  • 1,127
7 votes
1 answer
762 views

Feynman-Kac formula for the GFF

The Feynman-Kac formula says that $$ \exp(-t(-\Delta+V(X)))(x,y) = \mathbb{E}_{\gamma(0)=x,\gamma(t)=y}\left[\exp(-\int_0^t V\circ\gamma)\right] $$ where $\Delta$ is the Laplacian on $L^2(\mathbb{R}^n)...
PPR's user avatar
  • 396
7 votes
0 answers
269 views

Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$

I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by: $$ Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy $$ This operator ...
martin tassy's user avatar
7 votes
0 answers
549 views

Counter-example to the completeness of the Wasserstein metric

$\newcommand{\P}{\mathcal{P}}$ Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
Oleg's user avatar
  • 931
7 votes
0 answers
3k views

What is vague convergence and what does it accomplish?

For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
Greg Zitelli's user avatar
  • 1,124
7 votes
0 answers
299 views

Generalized Skorokhod spaces

Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
Tom LaGatta's user avatar
  • 8,512
7 votes
2 answers
1k views

Weighted Poincaré inequality

Consider a probability distribution $\pi$ with density $e^{-H(x)}$ on $\mathbb{R}$. Let us say that there is a Poincaré inequality with weight $w$ if for any smooth function $\phi$ satisfying $\int \...
Alekk's user avatar
  • 2,133
6 votes
2 answers
333 views

Is there a way to reconstruct the convolution $(f * g)(x)$ of $f$ with a Gaussian $g$ from sampled values, $(f*g)(a), a \in A$?

Suppose that $f: \mathbb{R} \to \mathbb{C}$ is a function which has support in $[-1,1]$. Let $g = g_\sigma$ be a centered Gaussian with variance $\sigma^2$. Is there a way to reconstruct the ...
J. Swail's user avatar
  • 437
6 votes
2 answers
979 views

Literature on behaviour of eigenfunctions under multiplication?

Dear community, I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable ...
herrsimon's user avatar
  • 199

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