All Questions
Tagged with measure-theory pr.probability
823 questions
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
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 ...
- 1,241
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
$$
P(X \in A) = P\left(X_n\in ...
- 479
6
votes
0
answers
388
views
Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
- 1,095
1
vote
1
answer
183
views
Diffuse measure space as a product of $[0;1]$ and another diffuse measure space
The title speaks of itself. How far is an arbitrary finite diffuse measure space from being almost isomorphic to a product of $[0;1]$ with another diffuse measure space? What would be reasonable ...
- 1,959
14
votes
1
answer
2k
views
Prokhorov's theorem in non separable metric spaces
Recently, working in some calculations I needed to use the Prokhorov's theorem
about compactness for probability measures. However, a friend warned me that
I had not the hypotesis of separability ...
- 757
7
votes
0
answers
3k
views
What is vague convergence and what does it accomplish?
For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
- 1,124
4
votes
1
answer
443
views
Uniform martingale convergence of Radon-Nikodym derivatives of a convex set of probabilities
Cross posted at MSE here. I'm hoping someone here can help complete zhoraster's answer. Any hints or references are appreciated.
Let $(\Omega, \mathcal{F})$ be a measurable space equipped with a ...
- 869
1
vote
0
answers
96
views
Infimum of equivalent measures
Suppose I have a functional of the form
$$
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
1
vote
1
answer
690
views
Integrable version of the Borel-Cantelli theorem?
I have taken an introductory course on measure theory where I learned about the Borel-Cantelli theorem but I wonder whether there is a lebesgue integrable version. Given an uncountable collection of ...
- 3,871
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 ...
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]) \...
- 4,466
3
votes
1
answer
940
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 $\...
- 8,071
3
votes
2
answers
923
views
On representing a continuous time Markov chain by a stochastic integral of a Poisson random measure
Let $Q=(q_{ij})$ be the transition rate matrix of a continuous time Markov chain $\{ X_t \}$ with countable state space $M$. Let $q_i = -q_{ii}=\sum_{j \neq i}q_{ij}$, and let $\Gamma_{ij}$ be defined ...
- 31
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 ...
- 319
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)$$
with a linear and continuous operator on a Banach space $A:X\to \mathbb R$ (in my case $X=C([0,1])$) and $\mu_0$ ...
- 375
1
vote
1
answer
165
views
Decomposition of $L^2$-spaces and singular measures
If $\langle \Omega, \mathfrak{F}, \mathbb{P}\rangle$ is a measure space and $L^2$ is the corresponding $L^2$ space and
$$
K\oplus K^{\perp} \cong L^2(\mathfrak{F},\mathbb{P}).
$$
Then let:
$$
\...
4
votes
3
answers
1k
views
Explicit example of second Borel–Cantelli lemma
Consider the probability model $(\Omega, \mathcal{F}, P)$ where $\Omega = [0,1]$, $\mathcal{F}$ is the Borel $\sigma$-algebra on $[0,1]$ and $P$ is the uniform measure on $[0,1]$.
Let $E_1, E_2, \...
- 487
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 $\mathcal{B}$. ...
- 349
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 $\...
- 111
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
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 ...
- 87
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, $Z$ ...
- 5,405
7
votes
1
answer
421
views
Convex representation of a measure
Let $\mathcal P(X)$ denote the space of all probability measure defined on a measurable space $X$. We canonically endow the former with its own measurability structure, generated by evaluation maps. ...
- 1,655
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 ...
- 675
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 ...
- 19
1
vote
0
answers
192
views
References about distances between singular probability measures
I would be interested in references on the topic of distances between probability measures that are singular with one another and not reduced to trivial ones. For example from here we know that total ...
- 1,334
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
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 ...
- 265
6
votes
1
answer
196
views
Simultaneous simulation of all probability measures on a compact metric space
A well known fact in probability is that a uniform random variable on $[0,1]$ can be used to simulate any other probability distribution on $\mathbb{R}$.
A standard way of doing this is to define, ...
- 4,304
3
votes
0
answers
428
views
When is the entropy of a $\sigma$-algebra finite?
Let two (countably-generated) $\sigma-$algebras $\mathscr{F,G}$ on the event space $\mathbb{R}$ be given. I believe we also need the atoms of $\mathscr{F,G}$ to be the points of $\mathbb{R}$.
Let $\...
- 2,680
1
vote
1
answer
148
views
Intuitional feeling of harmonic measure on one-third Cantor set
It is known that the harmonic measure on classical one-third Cantor set has Hausdorff dimension strictly less than $\frac{\log 2}{\log 3}$. Even harmonic measure has a close relation with brownian ...
- 1,706
3
votes
1
answer
734
views
Necessary and sufficient conditions for Kolmogorov's Extension Theorem
Let $(X_n,\mathcal{X}_n)$, $n=1,2,\ldots$ be measurable spaces. Define $Y_n = \prod_{k=1}^n X_k$ and let $\mathcal{Y}_n$ be the corresponding product $\sigma$-algebra. Similarly let $Y=\prod_{k=1}^\...
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\...
- 41
1
vote
1
answer
913
views
Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
- 247
6
votes
2
answers
735
views
Negative probabilities - what are two ordinary pgfs that correspond to the gf of a half-coin?
In Half of a Coin: Negative Probabilities, author considers pgf of a fair coin represented by random variable, $X = 1_H$:
$$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{...
- 247
2
votes
1
answer
200
views
Measurable isomorphism between two non-totally ergodic systems
Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for ...
- 319
4
votes
1
answer
203
views
Meaningful formalization of a continuum of Bernoulli random variables [closed]
I was wondering if there is a meaningful formalization for a continuum of Bernoulli random variables. Informally speaking, consider the interval $[0,1]$, and let's say that for every $x \in [0,1]$, ...
- 51
1
vote
1
answer
377
views
Order statistics of iid uniform RV and Pólya's urn model. Question about a.s. convergence
Let $U_1,U_2,U_3,\dots$ be IID uniform on $[0,1]$. For each $n\geq 1$ let
$$U_{1:n}<U_{2:n}<\dots<U_{n:n}$$
be the order statistic of $(U_1,\dots,U_n)$. Independent of the $U$ process there ...
- 265
10
votes
4
answers
792
views
Speed of convergence in Lebesgue's density theorem
Let $\lambda=\text{unif}([0,1])$ be uniform distribution on $[0,1]$ and $B$ be any Borel set. Lebesgue's density theorem states that for $\lambda$-almost all $x\in[0,1]$ the limit
$$\lim_{\epsilon\...
- 265
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 $x \in X$.
...
- 275
5
votes
1
answer
408
views
Conditions for existence of dominating $\sigma$-finite measure for all conditional distributions
Suppose $X$ and $Y$ are two real-valued random variables with a specified joint probability distribution $P_{X,Y}.$ I wish to determine if there is a $\sigma$-finite measure $\mu$ on the real line ...
- 1,269
2
votes
1
answer
358
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
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 ...
- 121
0
votes
1
answer
558
views
Counterexample: weak convergence doesn't imply $L^1-$convergence [closed]
I'm not sure my question is of research level, but I cannot find the answer in the existing reference. Let $\mu_n$ be a sequence of probability measures on $\mathbb R$ satisfying
$$\int_{\mathbb R}xd\...
- 1,835