All Questions
146 questions
-1
votes
1
answer
200
views
Is the unordered sum of measurable functions measurable?
Let $E$ be a normed $\mathbb R$-vector space and $I$ be a nonempty set. Remember that $(x_i)_{i\in I}\subseteq E$ is called summable if there is a $x\in E$ such that for all $\varepsilon>0$, there ...
- 167
3
votes
1
answer
265
views
Is the ball ratio theorem for Radon–Nikodým derivative known for general metric spaces?
Given two non-negative Borel measures $\mu$, $\nu$ on $\mathbb{R}^n$, that are finite on compact sets, such that $\nu\ll\mu$, it is well known that
$$\frac{d\nu}{d\mu}(x)= \lim_{\epsilon\to 0} \frac{\...
- 173
2
votes
0
answers
105
views
Equivalence between notions of dynamical coupling as defined by Villani in his book Optimal Transportation: Old and New
$\DeclareMathOperator\law{law}$In Villani's book he presents the following notions of dynamical couplings:
Let $(X,d)$ be a Polish space. A dynamical transference plan $\Pi$ is a probability measure ...
- 99
0
votes
1
answer
86
views
If $\lambda_i$ is symmetric with $\lambda_i\{0\}=0$, why does $\int_B1-\cos\langle x,x'\rangle\:(λ_1-λ_2)({\rm d}x)=0$ imply $λ_1=λ_2$?
Let $E$ be a separable $\mathbb R$-Banach space and $\lambda_i$ be a finite symmetric measure on $\mathcal B(E)$ with $\lambda_i(\{0\})=0$ and $$\int_B1-\cos\langle x,x'\rangle\:\underbrace{(\lambda_1-...
- 167
2
votes
1
answer
274
views
Small ball Gaussian probabilities with moving center
I would like to prove (if possible, otherwise find a counterexample for) the following lemma:
Let $(X,\|\cdot \|_X)$ be a separable Banach space. Additionally, we have a centred Gaussian measure $\mu$ ...
- 375
5
votes
2
answers
245
views
Differentiability of the map $x\mapsto \delta_x$ in the Arens-Eells/Lipschitz-free space
$\DeclareMathOperator\AE{AE}\DeclareMathOperator\Lip{Lip}$Let $\AE(X)$ denote the Arens-Eells space on a Banach space $X$. Consider the map:
$$
\begin{aligned}
\delta: X & \rightarrow \AE(X)
\\
x&...
0
votes
1
answer
248
views
Approximating arbitrary probability measures by discrete ones
Let $H$ be a separable Hilbert space and let $\mu$ be an arbitrary probability measure on $H$. I would like to approximate this distribution by by a finitely supported discrete distribution is the ...
- 2,288
2
votes
1
answer
173
views
Is there an analogue of transportation-cost inequality under a weighted Log-Sobolev Inequality?
It is known that under the Log-Sobolev Inequality for $\pi$, i.e., if for all $\rho$,
$$H_\pi(\rho):=\int \rho(x)\log\frac{\rho(x)}{\pi(x)}dx \leq \frac{1}{2\beta}\int \rho(x)\left\|\nabla \log\frac{\...
- 21
5
votes
1
answer
457
views
Sufficient condition for a probability measure to be a pushforward measure
Let $(E,d),(F,d')$ be separable metric spaces endowed with their Borel algebra, $f:E\rightarrow F$ a continuous surjective function, and $Q$ a probability measure on $F$ with separable support.
...
- 449
2
votes
0
answers
104
views
Weak convergence rates for integral operators
Suppose $q=\sum_{i=1}^m\pi_i\delta_{x_i}$ is a discrete measure on $\mathbb{R}^n$ and let $q\ast \varphi_\epsilon$ denote the convolution of $q$ with some mollifier $\varphi_\epsilon$, so that $q\ast\...
- 75
1
vote
0
answers
306
views
Gaussian measures on infinite dimensional spaces
On Zabczyk & Da Prato book about infinite dimensional SDEs they introduce the idea of Gaussian measures in infinite dimensional Banach spaces. They do so by means of Fernique theorem.
In the ...
- 515
1
vote
1
answer
137
views
Embeddings of spaces of probability measures
What is the relationship between the spaces $X_1\triangleq \mathscr{P}(C([0,1],\mathbb{R}))$ and $X_2\triangleq C([0,1],\mathscr{P}(\mathbb{R}))$; where $\mathscr{P}(\cdot)$ denotes the Borel ...
- 5,405
0
votes
0
answers
302
views
Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem
In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
- 167
4
votes
3
answers
2k
views
Duality of finite signed measures and bounded continuous functions
Let $E$ be a metric space, $C_b(E)$ denote the space of bounded continuous functions $E\to\mathbb R$ (equipped with the supremum norm), $\mathcal M(E)$ denote the space of finite signed measures on ...
- 167
2
votes
0
answers
142
views
Radon-Nikodým-like theorem for Radon measures
Let $(E,d)$ be a metric space, $\mu$ be a nonnegative Radon$^1$ measure on $\mathcal B(E)$ and $\nu$ be a finite (signed) Radon measure on $\mathcal B(E)$.
I'm searching for a Radon-Nikodým-like ...
- 167
2
votes
1
answer
183
views
Does set of finitely additive probability measures embed linearly into a strictly convex dual Banach space?
I am trying to better understand a condition that appears in Theorem 1 of this paper.
Let $K$ be a convex and compact subset of a locally convex tvs. The condition is:
$K$ embeds linearly into a ...
- 869
4
votes
0
answers
160
views
Can we show equivalence of two distributions based on their statistics?
Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \...
- 57
5
votes
1
answer
774
views
Question/References on the Skorokhod M1 topology
Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
user128095
2
votes
1
answer
122
views
How is this bound for a Wasserstein contraction coefficient in this paper obtained?
I'm trying to understand the following conclusion from this paper (see below for the relevant paragraphs):
I'm not sure whether they really mean that it follows from the statements of Lemma 3.2 (...
- 167
1
vote
0
answers
36
views
Does a total preorder on lotteries that preserves countable mixtures preserve arbitrary mixtures?
Let $X$ be a countable set. A lottery on $X$ is a function $\lambda: X \to [0,1]$ such that $\sum_x \lambda(x) = 1$. Let $\Delta X$ be the set of lotteries on $X$.
A total preorder $\preceq$ on $\...
- 869
12
votes
1
answer
1k
views
Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces
Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
- 333
2
votes
0
answers
520
views
Example of a non-reflexive Banach space and two sequences
Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$.
If $X$ is reflexive, ...
- 199
0
votes
1
answer
1k
views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
- 710
3
votes
1
answer
143
views
Density of $C(X,\operatorname{co}\{\delta_y\}_{y \in Y})$ in $C(X,\mathcal{P}(Y))$
Let $X,Y$ be locally-compact Polish spaces, equip the set $\mathcal{P}(Y)$ of probability measures on $Y$ with the weak$^{\star}$ topology (topology of convergence in distribution), and equip $C(X,\...
- 5,405
5
votes
1
answer
363
views
Inverse marginal property of a collection of $\sigma$-algebras
In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space"
I introduced the Inverse marginal property (IMP) for a collection of $\sigma$-algebras.
Let $(\...
- 935
0
votes
1
answer
102
views
Law of a step function and its generalization to two dimensions on an appropriate spaces
Let's consider two discontinuous functions defined on $D$ and $D \times [0,T]$, respectively:
A step function: $u_1(x)=\begin{cases}
u_{L}, x<c_1, \\[2ex]
u_{R}, x>c_1,
\end{cases}$
A "...
- 657
1
vote
0
answers
169
views
A question about Stroock's notes on the Weyl lemma
On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
- 1,461
1
vote
0
answers
83
views
Embedding random variables in infinite-dimensional spaces
Let $H$ be a reproducing kernel Hilbert space of functions $f:E\to F$ with kernel $k$. A point in $E$ may be embedded into $H$ via the canonical embedding $x\mapsto k(x,\cdot)$. Similarly, a random ...
- 710
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*...
- 173
0
votes
1
answer
133
views
Product of sets with the Radon-Nikodym Property (RNP)
I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP.
Does the above result ...
- 1,245
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 ...
- 710
6
votes
1
answer
575
views
Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel
I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant.
Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
- 1,495
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 ...
- 243
3
votes
2
answers
926
views
Weak convergence of conditional probabilities
Suppose $\mu_n\implies\mu$, i.e. $\mu_n$ converges weakly to $\mu$ where $\mu_n$, $\mu$ are probability measures on some metric space $(X,d)$. Given a Borel set $B$, define $\mu^B$ to be the ...
- 710
2
votes
1
answer
2k
views
Convergence of probability density function
There are various kinds of (convergence of random variables) but I have never read about convergence of density functions.
Let $X_1, X_2, \dots, X$ be random variables $\Omega \to \mathbb{R}$ and $...
- 23
1
vote
0
answers
58
views
Extension of a result about measurable, additive functionals
Let $W$ be a set, and let $v$ be a finitely additive probability measure on $2^W$.
Equip $2^W$ with the Borel sigma-algebra $\mathcal{B}$ generated by the sub-basic sets of the form $\{a: w \in a\}$ ...
- 869
4
votes
1
answer
412
views
Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator?
I have a basic question about Gaussian measures on a Hilbert space:
Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the ...
- 617
2
votes
1
answer
263
views
Schwartz space on $\bigcup_{n=1}^CR^n$
I have an application where I need to work with the following idea.
Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...
- 21
5
votes
2
answers
470
views
Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$
Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...
- 1,245
2
votes
1
answer
203
views
Non-uniqueness in Krylov-Bogoliubov theorem
So apparently the Krylov-Bogoliubov theorem says that every continuous function $f:X\to X$ on a compact metrizable space $X$ has an invariant probability measure $\mu$.
Of course, if $X$ is just a ...
- 24.8k
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) ...
- 931
6
votes
0
answers
388
views
Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
- 1,095
1
vote
1
answer
183
views
Diffuse measure space as a product of $[0;1]$ and another diffuse measure space
The title speaks of itself. How far is an arbitrary finite diffuse measure space from being almost isomorphic to a product of $[0;1]$ with another diffuse measure space? What would be reasonable ...
- 1,959
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 ...
- 1,124
1
vote
0
answers
96
views
Infimum of equivalent measures
Suppose I have a functional of the form
$$
F(\mathbb{P})\triangleq \int_{\mathbb{R}^d} \int_{\Omega}f(x,\omega)\mathbb{P}(d\omega)m(dx),
$$
where $m$ is the Lebesgue measure and $\mathbb{P}$ is a ...
- 173
9
votes
2
answers
548
views
What mode of convergence is this?
I'm interested in a new (to me) mode of convergence which is stronger than convergence in measure/probability. I want to know if it has a name and if it is used much in the literature. I will write ...
- 6,287
2
votes
1
answer
115
views
Normalization of Gaussian w.r.t. Gaussian in a Banach space
I would like to compute
$$\int_X \exp\left(-\frac{1}{2}(Au)^2\right)\mathrm d\mu_0(u)$$
with a linear and continuous operator on a Banach space $A:X\to \mathbb R$ (in my case $X=C([0,1])$) and $\mu_0$ ...
- 375
1
vote
1
answer
165
views
Decomposition of $L^2$-spaces and singular measures
If $\langle \Omega, \mathfrak{F}, \mathbb{P}\rangle$ is a measure space and $L^2$ is the corresponding $L^2$ space and
$$
K\oplus K^{\perp} \cong L^2(\mathfrak{F},\mathbb{P}).
$$
Then let:
$$
\...
2
votes
1
answer
357
views
Measurability of integrals with respect to different measures
Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
- 405
2
votes
0
answers
168
views
Interchanging integrals and continuous linear forms in RKHS
I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan.
In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the ...
- 619