All Questions
Tagged with pr.probability measure-theory
823 questions
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
3
votes
1
answer
732
views
What does $\pi$ in the term $\pi$-system stand for?
In measure theory, what does the $\pi$ in $\pi$-system stand for? Also, what about the $\lambda$ in $\lambda$-system? I want to know why Dynkin chosen these names, and why these names make sense.
- 8,071
6
votes
2
answers
701
views
Wiener Measure measure on functions?
I know that the Wiener measure for the Brownian motion $\{B_t\}_{t\ge 0}$ on the probability space $(\Omega, \mathscr{F},P)$ can be defined as $\mu=P\circ B^{-1}$ acting on the sigma-algebra generated ...
- 8,071
1
vote
1
answer
649
views
Extreme Points of a set of distributions with moment and/or support constraint
Let $X$ be a random variable with the distribution $F$ (cdf).
What are the extreme points of the sets of the form:
\begin{align}
P_1&=\left\{ F: \int |x|^k dF\le c \right\},\\
P_2&=\left\{ F:...
- 17.2k
20
votes
1
answer
2k
views
Does every compact metric space have a canonical probability measure?
Edit: Shortly after this post it was rightly pointed out by @AntonPetrunin that the measure $\mu$ may not be unique. @R W then showed how one can construct a metric space where the limiting measure is ...
CommunityBot
- 1
3
votes
3
answers
656
views
Free probability with unbounded random variables?
This is partially inspired by this question and this blog post.
When trying to express classical probability in the "free probability" setting one takes an algebra of random variables equipped with ...
- 301
1
vote
1
answer
201
views
Measure of bounded fourth (and below) moment distributions?
Many results in probability theory/random matrix theory/etc require probability distributions with finite fourth moments; what is the measure of such probability distributions (in the space of ...
- 29.8k
4
votes
1
answer
352
views
Measure of the rate of convergence for filtration and conditional expectations
This question is cross-posted at MSE with a soon to expire bounty that hasn't generated much discussion.
Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_n)_n$ a filtration that ...
- 8,071
2
votes
0
answers
130
views
A question on probability measure on the unit ball of Banach spaces
Let $X$ be a Banach space and let $(x^{*}_{n})_{n}$ be a sequence in $X^{*}$. Suppose that $\sum_{n}|\langle x^{*}_{n},x\rangle |\leq \|x\|$ for all $x\in X$.
Question: Is there a probability measure ...
- 24.2k
3
votes
1
answer
156
views
Measurability of a particular set generated by discrete probability measures
Suppose that $(S,\Sigma)$ is a measurable space with $S$ Polish and $\Sigma$ its Borel sigma algebra. Let $\mathcal{C}$ be the collection of discrete probability measures on $S$ having countably ...
- 23.2k
1
vote
1
answer
142
views
Nonrandomized probability kernels
I've asked this question also on mathematics stackexchange, but despite nearly two dozen views, there isn't a single comment, nevermind an answer. Any help would be appreciated.
Update: See update 1 ...
CommunityBot
- 1
4
votes
0
answers
358
views
Lipschitz kernel
We consider the following probability measure on $\mathbb{R}^2$:
$\mu = Leb\vert_{[0,1]} \times \delta_0$. Furthermore the following dilation, say $d$, is defined as $(x,0) \mapsto \frac{1}{2}(\delta_{...
- 141
2
votes
1
answer
254
views
Measurability of a parametrized conditional expectation
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\mathcal{G}\subset\mathcal{F}$ a Sub-$\sigma$-Algebra. Moreover, let $X:\Omega\rightarrow\mathbb{R}$ be a random variable and $F:\...
- 169
18
votes
3
answers
2k
views
How do we express measurable spaces using type theory?
A measurable space $(X,\mathcal X)$ consists of a set $X$ equipped with a $\sigma$-algebra of subsets $\mathcal X$. I would like to write computer programs involving measurable spaces, but to the best ...
CommunityBot
- 1
3
votes
1
answer
941
views
What is the mathematical characterization of sufficient statistics of a given $\sigma$-dominated probability model?
Given a probability model $\mathcal{P}=\{P_{\theta},\theta \in \Theta \}$ dominated by a $\sigma$-finite measure $\lambda$ (e.g. Lebesgue measure) on a locally compact space $\cal{X}$ along with $\...
CommunityBot
- 1
13
votes
4
answers
5k
views
What is known about the Gaussian measure of the unit ball in a Hilbert Space?
Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$. What do we know about $\mu(B(0,1))$, where $B(0,1)$ ...
- 25.5k
2
votes
1
answer
217
views
Measure space for trees and other algebraic datatypes
Given a measure space $\mathcal M$, I am wondering what kind of measure space $\mathcal T(\mathcal M)$ one could associate to the set of binary trees with elements from $\mathcal M$ at each node.
The ...
- 8,071
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 $E$ ...
- 8,071
4
votes
1
answer
1k
views
For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?
I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
CommunityBot
- 1
4
votes
1
answer
227
views
Event of positive probability occurs infinitely often in stationary ergodic sequence
Setup:
Suppose $X = \{X_n\}_{n\in\mathbb{Z}}$ is a stationary ergodic proces on the real line and let $A = \prod_{n\in\mathbb{Z}}A_n$ be a Borel measurable set such that
$$
P(X \in A) = P\left(X_n\in ...
- 18.7k
1
vote
1
answer
183
views
Diffuse measure space as a product of $[0;1]$ and another diffuse measure space
The title speaks of itself. How far is an arbitrary finite diffuse measure space from being almost isomorphic to a product of $[0;1]$ with another diffuse measure space? What would be reasonable ...
4
votes
1
answer
443
views
Uniform martingale convergence of Radon-Nikodym derivatives of a convex set of probabilities
Cross posted at MSE here. I'm hoping someone here can help complete zhoraster's answer. Any hints or references are appreciated.
Let $(\Omega, \mathcal{F})$ be a measurable space equipped with a ...
CommunityBot
- 1
14
votes
1
answer
2k
views
Prokhorov's theorem in non separable metric spaces
Recently, working in some calculations I needed to use the Prokhorov's theorem
about compactness for probability measures. However, a friend warned me that
I had not the hypotesis of separability ...
- 10.3k
7
votes
0
answers
3k
views
What is vague convergence and what does it accomplish?
For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
CommunityBot
- 1
11
votes
2
answers
2k
views
Multi-dimensional moment problem
Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment
$$ m_{\bar i} = \int x_i^{i_1} x_2^{i_2}...
- 3,574
11
votes
2
answers
466
views
Defining measures over frames in place of $\sigma$-algebras
Normally, measures and probability spaces are defined over $\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of $\sigma$-algebras? Specifically, ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
2
votes
0
answers
924
views
Isomorphism of probability spaces
Consider a surjective map $f:(X, \sigma(f)) \to (Y, \mathcal{Y})$, if a measure $\nu$ is given in $(Y, \mathcal{Y})$ the pullback $\nu(f(\cdot))$ is a measure on $(X, \sigma(f))$, similarly if $\mu$ ...
- 349
2
votes
1
answer
251
views
Automorphism on the unit interval compatible with a measure preserving set function
Cross-posting from math stack-exchange since it's not getting any visibility there.
I am given a function $F: \{[0, y]: y \in I\} \to \Sigma(I)$, such that $\lambda(F([0, y])) = y$, and $F([0, y]) \...
- 18.7k
3
votes
0
answers
509
views
sufficient condition for the continuity of conditional probability wrt the conditioning variable
Given a regular conditional probability $P(X\in B | T(X) = t)$, where $T$ is a continuous mapping from $\mathcal{X}$ (on which $X$ is defined) to $\mathcal{T}$. Do we know any sufficient condition for ...
CommunityBot
- 1
2
votes
0
answers
103
views
measures in infinite dimension space of entire functions [closed]
It is known that there is no canonical generalization of Lebesgue measure in infinite dimension of function spaces. Since it seems that the space of (transcendental) entire function seems improtant ...
- 1,706
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 $\mu_0$ ...
- 34.7k
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:
$$
\...
- 29.8k
2
votes
1
answer
553
views
Continuity sets as generator of the $\sigma$-algebra generated by cylinders
On $(\mathbb{R}, \mathcal{B})$ given any finite measure $\mu$ the sets of the form (continuity sets) $$\{A \in \mathcal{B} : \mu(\partial A) = 0\}$$ generate the Borel $\sigma$-algebra $\mathcal{B}$. ...
CommunityBot
- 1
2
votes
4
answers
610
views
How to generalize normal number theorem
The Borel number theorem states that with respect to Lebesgue measure, almost all real numbers are normal numbers. It is sometimes stated in the context of the compact interval $[0,1]$, where one ...
CommunityBot
- 1
1
vote
0
answers
188
views
Regular measure in finite Borel sets [closed]
I have a question concerning
these lecture notes, https://www.math.leidenuniv.nl/~vangaans/jancol1.pdf
In the proof of the proposition 2.3 (page 3), there are two steps:
1) define the family $\...
- 6,045
4
votes
0
answers
95
views
Approximating martingales given marginal distributions
Let $(\mu_0,\mu_1)$ be a vector of probability measures on $\mathbb R$ that are of finite first moment, i.e.
$$\int_{\mathbb{R}}|x|\mu_i(dx)~<~+\infty \mbox{ for } i=0,1$$
and increasing in ...
- 1,835
2
votes
0
answers
63
views
Sensitivity of a function against its random arguments
Let $g:R^{n+m} \to R$ be a deterministic function of some independent random variables $x_1,\ldots,x_n$ with distributions $f_{x_1}(x),\ldots,f_{x_n}(x)$ and some deterministic variables $z_1,\ldots,...
- 482
5
votes
0
answers
178
views
Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?
Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mathscr{P}$ be a probability ...
- 2,306
0
votes
1
answer
172
views
Taking away the "almost sure" [closed]
Given an arbitrary sequence of random variables (or say measurable functions on a finite-measure space) $\xi_n$, one can show by a truncation and Borel-Cantelli argument that there always exists a ...
- 5,169
6
votes
1
answer
188
views
Does there exist a Penalized Conditional Expectation?
In my recent work I've become interested in working with the minimizer of
$$
\mathbb{E}[(Y-Z)^2] + \lambda P(Z),
$$
$Y$ is an observed random variable, $P$ is a positive-convex penalty function, $Z$ ...
- 4,529
4
votes
1
answer
721
views
Conditions for supremum and conditional Expectation to commute
I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and have (for simplicity) two stopping times $\tau_1$ and $\tau_2$ such that $\tau_2 \leq \tau_1$ and $U:\Bbb R\...
CommunityBot
- 1
9
votes
2
answers
2k
views
common dominating measure for a family of measures
Given a family $\{\mu \}_{i\in I}$ on a Polish space (complete, separable metric space) $X$. When does there exist a measure $\lambda$ such that
$$\mu_i=f_i \lambda$$
where the $f_i$ are densities (...
- 4,034
19
votes
3
answers
3k
views
Measure induced on [0, 1] by infinite tosses of biased coin
It is well-known that one can get the Lebesgue measure on [0, 1] by tossing a fair coin infinitely (countably) many times and mapping each sequence to a real number written out in binary.
I was ...
CommunityBot
- 1
1
vote
1
answer
510
views
Total variation distance between multinomial laws
Can someone help me with the following problem:
Let $P_n$ and $Q_n$ two multinomial laws with parameters $(p,n)$ and $(q,n)$, where $p$ and $q$ are two probability measures on some measurable space ...
- 1
5
votes
1
answer
209
views
Measurable $\epsilon$-optimal selection with an analytically measurable stochastic kernel
Let $(X, \mathcal{X})$ and $(A, \mathcal{A})$ be standard Borel spaces, $D \subseteq X \times A$ be an analytic set, and $D_x := \{a \in A : (x, a) \in D\}$ denote the $x$-section of $D$ at $x \in X$.
...
CommunityBot
- 1
2
votes
0
answers
160
views
A construction of abstract Wiener spaces using Prokhorov's theorem
I am struggling with Leonard Gross's (original) construction of abstract Wiener spaces (AWS). His proof is somewhat convoluted, but from what I have been able to understand he constructs a certain ...
- 5,407
1
vote
1
answer
377
views
Order statistics of iid uniform RV and Pólya's urn model. Question about a.s. convergence
Let $U_1,U_2,U_3,\dots$ be IID uniform on $[0,1]$. For each $n\geq 1$ let
$$U_{1:n}<U_{2:n}<\dots<U_{n:n}$$
be the order statistic of $(U_1,\dots,U_n)$. Independent of the $U$ process there ...
- 61.9k
4
votes
1
answer
331
views
Lebesgue Density Theorem: From convergence in probability to a.s. convergence
Let $\Omega_1,\Omega_2,\dots$ be a sequence of finite nonempty sets endowed with discrete topology. The product space
$$\Omega:=\Omega_1\times \Omega_2\times\cdots=\prod_{n\geq 1}\Omega_n$$ can be ...
CommunityBot
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