All Questions
Tagged with pr.probability measure-theory
823 questions
2
votes
3
answers
458
views
More natural example of measurable but not progressive process
All examples of measurable but not progressive processes I have ever seen seemed to be based on the huge difference between $\mathcal{F}$ and $\mathcal{F}_\infty$. Here is what I mean.
Consider ...
- 620
6
votes
2
answers
173
views
When can we extend a function on a $\lambda$-system to a probability measure?
Let $\Omega$ be a nonempty set and let $\mathcal{L}$ a $\lambda$-system on $\Omega$. That is,
(i) $\Omega \in \mathcal{L}$,
(ii) if $A, B \in \mathcal{L}$ and $A \subseteq B$, then $B \setminus A \in \...
- 128k
1
vote
1
answer
422
views
Motivation for Ionescu-Tulcea extension theorem (as opposed to Kolmogorov's)
I recently asked a question on the differences between Ionescu-Tulcea and Kolmogorov extension theorems (ITET and KET for short). A lot of my confusion has been cleared there and what I understood ...
- 620
0
votes
1
answer
77
views
Meyer's example of a separable process with no path regularity
This question is a cross-post from math.stackexchange.com. I am reposting it here since I didn't receive an answer there. The original post can be found by this link.
In the following excerpt from ...
- 128k
6
votes
2
answers
756
views
Kolmogorov vs Ionescu-Tulcea extension theorem (again)
Disclaimer. This post is not a duplicate, I have carefully (best I could) read all posts on the subject both here and on math.se and my particular questions have not been asked there.
I've recently ...
- 1,989
1
vote
1
answer
62
views
Let $c: X \times Y \to \overline{\mathbb R}$ be $\gamma$-measurable. Is $c_x:Y \to \overline{\mathbb R}, y \mapsto c(x, y)$ $\nu$-measurable?
Let $(X, \mathcal X, \mu)$ and $(Y, \mathcal Y, \nu)$ be $\sigma$-finite measure spaces. Let $\overline{\mathbb R} := \mathbb R \cup \{\pm \infty\}$.
$f:X \to \overline{\mathbb R}$ is called $\mu$-...
0
votes
1
answer
96
views
What is the significance of Blumenthal and Getoor's result on the boundedness of paths of a standard Markov process?
In the book Markov processes and Potential Theory of Blumenthal and Getoor we can find the following result:
I don't understand the significance of this result. If I don't misinterpret the assertion, ...
CommunityBot
- 1
9
votes
1
answer
1k
views
Can a non-Borel set be a standard Borel space?
Recall that a standard Borel space is a measurable space $(X,\mathcal{M})$ (i.e. a set with a $\sigma$-algebra) such that there exists a 1-1 bimeasurable map $\phi$ from $(X,\mathcal{M})$ to $[0,1]$ (...
- 4,713
3
votes
2
answers
277
views
Is there something like a "self-avoiding Markov chain" on a continuous space?
If stumbled accross self-avoiding walks. They seem to be deterministically generated, but from a quick google search term seem to be randomly generated variants.
However, as far as I can see they are ...
- 8,992
1
vote
0
answers
43
views
Does the constrained Wasserstein barycenter admit a blue noise property?
Let $(E,d)$ be a metric space and $\nu$ be a probability measure on $\mathcal B(E)$. In this paper, it is mentioned that sampling from $\mu$ can be described as choosing $n\in\mathbb N$, $x_1,\ldots,...
- 167
11
votes
3
answers
565
views
Is Stoch enriched in Met?
Let $\mathsf{Stoch}$ denote the Kleisli category of the Giry monad. That is, $\mathsf{Stoch}$ is a category whose objects are measurable spaces and for which a morphism $f\in\mathsf{Stoch}(X,Y)$ is a ...
- 399
2
votes
0
answers
150
views
A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$
Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \...
- 657
25
votes
6
answers
6k
views
Proof of Krylov-Bogoliubov theorem
Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
- 4,713
1
vote
0
answers
166
views
Wiener Integral and its distribution
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space.
Let $(W(t))_{x \in \mathbb{R}^d}$ be a Gaussian random field.
Then, we can define Wiener integral $\int_{\mathbb{R}^d} f(\xi) \, dW(\xi)$...
2
votes
0
answers
92
views
A variant of disintegration theorem where the assumptions on $f$ and $g$ are exchanged
I have recently read about about disintegration theorem, i.e.,
Disintegration theorem Let
$X$ be a Polish space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X$...
- 835
1
vote
2
answers
140
views
On conditions for the existence of $h\in L^1$ such that $h>0$ a.e
I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has the following proposition.
If $\mu$ is $\sigma$-finite, then there is a function $h \in L^{1}(\mu)...
- 16.4k
2
votes
1
answer
201
views
Joint irreducibility and aperiodicity of two independent Markov chains
Let $(X_i)_i, (Y_i)_i$ be two independent Markov chains on Polish state spaces $\mathcal{X}, \mathcal{Y}$ and with kernels $P, Q$. Suppose that they are both $\psi$-irreducible and aperiodic and have ...
- 128k
2
votes
1
answer
806
views
A question about the proof of the Levy-Khintchine representation Theorem
I'm studying Infinitely Divisible random variables using this Lecture Notes. And I have a question that is driving me crazy.
In the proof of the "only if" part of the Levy-Khintchine ...
- 128k
5
votes
0
answers
266
views
Concentration inequalities for random measures
For random variables $X_1,\dots,X_n$ with common mean $\mathbb{E}[X_i]=\mu$ and common bounds $a\leq X_i\leq b$, we have the very useful Hoeffding's inequality:
$$\mathbb{P}\left(\left|\mu -\frac1n\...
- 101
4
votes
2
answers
374
views
Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation
I'm reading a proof of below theorem from this paper.
Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
- 271
10
votes
1
answer
797
views
Proof of Lomnicki and Ulam on infinite product probability spaces
Given an arbitrary, nonempty family $(\Omega_i,\Sigma_i,\mu_i)_{i\in I}$ of probability spaces, there exists a probability measure $\mu$ on $\otimes_i\Sigma_i$ such that for every finite set $F\...
- 63.9k
2
votes
2
answers
943
views
measuring distance between probability measures only at the tail
Is there any official (i.e., to be found in probability books) metric for the distance between two probability measures, defined only on a subset of their support?
Take, for example, the total ...
- 2,821
32
votes
4
answers
4k
views
Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
- 4,713
4
votes
0
answers
166
views
Does the existence of regular conditional measure follow from that of regular conditional distribution?
For more succinct description, I use the following abbreviations, i.e.,
RCPD: Regular conditional probability distribution.
RCPM: Regular conditional probability measure.
First are definitions 10.4....
- 657
4
votes
1
answer
439
views
Push-forward of uniform measure and uniqueness
On a standard Borel (or Polish) space $X$, any probability measure $\mu$ is the push-forward of the uniform measure on $[0, 1]$ under some $f : [0, 1] \to X$.
This $f$ is not unique in general. ...
- 10.3k
4
votes
0
answers
492
views
Disintegration of measures: a confusion about an existence proof from a lecture note
I'm reading a proof of Theorem 2.25 below from this note. First, we recall a definition and a theorem, i.e.,
Theorem 2.25 (Disintegration). Let $\left(Z, d_Z\right)$ and $\left(X, d_X\right)$ be ...
- 657
3
votes
0
answers
175
views
Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?
I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
- 204
5
votes
0
answers
135
views
Criteria for tightness of Gaussian measures on Banach spaces
In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...
- 505
3
votes
1
answer
190
views
Is the function $x\mapsto(\Delta x(t))_{t\ge0}$ measurable with respect to the product $\sigma$-algebra?
Let $E$ be a normed $\mathbb R$-vector space. If $x:[0,\infty)\to E$ is càdlàg, let $$x(t-):=\lim_{s\to t-}x(s)\;\;\;\text{for }t\ge0$$ ($x(0-):=0$) and $$\Delta x(t):=x(t)-x(t-)\;\;\;\text{for }t\ge0....
CommunityBot
- 1
2
votes
1
answer
95
views
Compactness of the integral of a set-valued function
Let $X$ be a compact space (e.g. a compact subset of $\mathbb R^n$) and $P$ be a probability measure on $X$. Let $A$ be a compact subset of some $\mathbb R^d$. Finally, let $F$ be the collection of $P$...
CommunityBot
- 1
5
votes
1
answer
430
views
Volume of a shape whose boundary consists of portions of spheres symmetrically placed about the origin in $d\gg 1$ dimensions
We are given a convex shape $S$ in the $d$-dimensional Euclidean space, whose boundary is formed by portions of $2d$ different spheres, one portion per sphere. The radius of each sphere is the same, $...
- 1,677
2
votes
1
answer
197
views
$\Psi$ in finite Wiener–Itô Chaos implies existence of continuous representative on neighborhood of Cameron–Martin space?
Let $(\Theta, H, \mu)$ be an abstract Wiener space, i.e. let $(\Theta, \lVert \cdot \rVert_{\Theta})$ be a separable Banach space, let $(H, \langle \cdot, \cdot \rangle_{H})$ be a separable Hilbert ...
- 10.3k
0
votes
1
answer
105
views
Transforming two smooth densities to the same density
I am looking for an example of the following:
Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the ...
- 716
1
vote
0
answers
177
views
Building random homeomorphisms of the torus $\mathbb T^2$
In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
- 101
8
votes
2
answers
1k
views
Kolmogorov 0-1 law counter examples for almost independent variables
According to Kolmogorov 0-1 law, if $ \left(X_{i}\right)_{i=1}^{\infty} $ are independent, then the tail sigma algebra is trivial. I want to construct such variables which are "almost independent&...
- 148k
1
vote
1
answer
185
views
Cameron-Martin space of product space
Suppose you have Banach spaces $\mathcal B_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu_\alpha$ be Gaussian measures on $\mathcal B_\alpha$ with Cameron-Martin spaces $\mathcal H_{\mu_\...
- 150
0
votes
0
answers
118
views
A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
- 101
1
vote
0
answers
96
views
Building random homeomorphisms of the circle
Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as
\...
- 101
9
votes
1
answer
372
views
Why impossible events have some drawbacks or pathologies in probability theory?
It is said by Halmos, P.R.; in "Lectures on ergodic theory"
"Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure ...
- 82.7k
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 ...
- 128k
2
votes
1
answer
373
views
Bakry-Emery criterion
The most common use of the Bakry-Emery criterion is for the measure $\mu(x)=e^{-u(x)} /Z$ where $u \in \mathcal{C}^2$. I would like to ask for an application to a smaller class.
Consider $u(x)=|x|^2 + ...
- 1,081
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
$$X_1^{q} = (\times_{i=1}^q\...
- 6,045
3
votes
3
answers
5k
views
Hoeffding's inequality for vector valued random variables
Is there a version of Hoeffding's inequality for vector valued random variables?
This seems to be hard to find and I wonder why. I suppose it is difficult to show Hoeffding's lemma, since the proof ...
- 111
1
vote
0
answers
37
views
If $(\kappa_t)_{t\ge0}$ is a Markov semigroup with invariant measure $μ$, under which assumption is $t\mapsto\kappa_tf$ measurable for $f\in L^p(μ)$?
Let
$(E,\mathcal E)$ be a measurable space;
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$;
$\mu$ be a finite measure on $(E,\mathcal E)$ which is subinvariant with respect to $(\...
- 167
7
votes
1
answer
253
views
Are all quasi-regular points on Polish spaces generic points?
Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
- 1,074
-1
votes
1
answer
273
views
What does $\mu$ and $\nu$ "dependent" mean? [closed]
On the other hand, if $\mu$ and $\nu$ are completely dependent then $\pi_{x_1}=\delta_{f(x_1)}$ for some function $f$. Then $W(\pi)=1$.
Note that
$$
\pi(dx, dy)=\pi_x(dy)\mu(dx).
$$
If $\pi_x(dy)=\...
- 288
0
votes
1
answer
262
views
Construction of a Markov process with prescribed local behavior and state-dependent jump distribution
Let
$(E,\mathcal E)$ be a measurable space
$\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\...
- 167
2
votes
1
answer
122
views
Can we say that there exists a measurable function $f$ such that $ \nu=f_{\#}\mu$?
Define a coupling $\pi\in \Pi(\mu,\nu)$ on the product space $(X\times X,\mathcal{F}\times\mathcal{F})$. let $\pi_x$ be the disintegration of $\pi$ with respect to the $\mu$, i.e. there exists a Borel ...
- 288
0
votes
1
answer
284
views
Is there a "smooth Kantorovich-Rubinstein duality" for Wasserstein distances on smooth/Euclidean space?
Let $X$ be a compact metric space, and fix an arbitrary point $x_\ast \in X$. By the Kantorovich-Rubinstein duality theorem, the $1$-Wasserstein metric $W_1$ on the set of Borel probability measures ...
- 128k
5
votes
1
answer
254
views
Is the topology of weak+Hausdorff convergence Polish?
Let $X$ be a compact metric space, $P_X$ the set of Borel probability measures on $X$, and $K_X$ the set of non-empty closed subsets of $X$. I will define the "topology of weak+Hausdorff ...
- 41.9k