All Questions
Tagged with pr.probability measure-theory
823 questions
3
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428
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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
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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 ...
- 66.3k
1
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0
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227
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Uniform Law Of Iterated Logarithm for VC classes
Kenneth Alexander proved a uniform Law Of Iterated logarithm for Vapnik-Chervonenkis classes in the article Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm (Ann. ...
- 3,968
3
votes
1
answer
734
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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}^\...
- 1,074
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2
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735
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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, $X = 1_H$:
$$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{...
- 247
6
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1
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1k
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About the generating structure of Borel field
This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer.
On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...
CommunityBot
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2
votes
1
answer
200
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Measurable isomorphism between two non-totally ergodic systems
Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for ...
- 3,911
4
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1
answer
203
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Meaningful formalization of a continuum of Bernoulli random variables [closed]
I was wondering if there is a meaningful formalization for a continuum of Bernoulli random variables. Informally speaking, consider the interval $[0,1]$, and let's say that for every $x \in [0,1]$, ...
- 265
10
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4
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792
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Speed of convergence in Lebesgue's density theorem
Let $\lambda=\text{unif}([0,1])$ be uniform distribution on $[0,1]$ and $B$ be any Borel set. Lebesgue's density theorem states that for $\lambda$-almost all $x\in[0,1]$ the limit
$$\lim_{\epsilon\...
CommunityBot
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4
votes
2
answers
2k
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Change of time or change of measure
Consider simple diffusion $dX_t = \sigma dw_t$ and a parameter $a>0$ and $X_0=x$. Let us denote $Y_t = X_{at}$ - thus we made a change of time. Let us denote an original measure as $P$. How to find ...
- 1,334
3
votes
2
answers
331
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Extreme couplings
Let $X,Y$ be Polish spaces, and $\mu$ and $\nu$ are probability measures on $X$ and $Y$ respectively. We say that $M$ is a coupling of $\mu$ and $\nu$ if it is a probability measure on $X\times Y$, ...
- 8,501
14
votes
1
answer
2k
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surprisingly difficult filtration problem
I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky:
Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
CommunityBot
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5
votes
1
answer
408
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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 ...
- 1,989
2
votes
1
answer
358
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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 ...
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1
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558
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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\...
- 128k
2
votes
1
answer
363
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Integration against Borel measures on compact Hausdorff spaces
I am studying the properties of integration against Borel measures and Baire measures. And I am not sure whether the following proposition is correct and I tried to give a proof.
Suppose that $X$ ...
- 128k
7
votes
1
answer
621
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Does every (generalized?) Markov chain admit transition probabilities?
To pose the question let us start by recalling the following notions:
Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and $(V,\mathcal{V})$ ...
- 486
1
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0
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364
views
Bounds on Wasserstein (Kantorovich) distance
Let $X$ be a Polish space endowed with a bounded metric $\rho_X$. Let $\mu, \mu'$ be two probability measures, and $\kappa, \kappa'$ be two stochastic kernels on $X$. Assume that $\kappa, \kappa'$ are ...
- 1,655
2
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1
answer
144
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Do we have independence if we let the indices of the events increase?
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
$A_{m,1}...
CommunityBot
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2
votes
0
answers
168
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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
2
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1
answer
560
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Covariance matrix as optimization problem solution?
I have seen the expectation of a random vector expressed as the solution to the optimization problem:
\begin{equation}
\mathbb{E}[X]=argmin_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= \int_{\...
- 5,405
4
votes
3
answers
713
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Measure of intersections in probability spaces
Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$.
...
- 37.7k
2
votes
1
answer
202
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What is the formal name of this set-related concept?
I "invented" a concept and it feels like it has already been invented before. I would like to know whether such a concept exists and if so, what is its name?
Let $S$ be a family of finite sets.
Say ...
- 3,549
1
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0
answers
139
views
weak-* versus entropy growth
General question. Let $\eta_{n}$ be a sequence of invariant measures on $\{0,1,2,...,p-1\}^{\mathbb{N}}$ and $B$ the Bernoulli uniform measure. Knowing that $\eta_{n} \rightarrow B$ in the weak-* ...
- 105k
6
votes
1
answer
930
views
Weak convergence in random measures
I don't understand the following as I read along a proof in a paper (Page 66, "Asymptotic Behaviour of some interacting systems", by Sylvie Meleard):
We denote by $\mathcal{P}({M})$ the ...
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1
answer
557
views
Is the limsup or liminf of n-wise independent events independent?
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
$A_{m,1}...
CommunityBot
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3
votes
1
answer
1k
views
If $H$ is a separable Hilbert space, is its dual dense in $L^2(H)$?
Let $H$ be an infinite-dimensional, separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Let $H^*$ denote the space ...
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1
vote
1
answer
166
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Question abouth Skorokhod representation of random variables (II)
This is a continuation of
Question abouth Skorokhod representation of random variables
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that
$$\int_{\mathbb R}|x|^pd\mu(x),~ \...
CommunityBot
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4
votes
1
answer
220
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Question about the weak convergence of probability
Let $\mu$ be a probability measure on $\mathbb R$ and set
$$c(K):=\int_{\mathbb R}(x-K)^+d\mu(x).$$
Assume that one has a sequence of probability measures $(\mu_n)_{n\ge 1}$ s.t.
$$\int_{\mathbb R}\...
CommunityBot
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5
votes
1
answer
356
views
Question abouth Prokhorov metric
Let $X$ and $Y$ be two random variables with first order moments, i.e. $E[|X|]$, $E[|Y|]<+\infty$. Assume further that
$$E\left[|X-Y|\right]<\varepsilon.$$
Set $Law(X)=\mu$ and $Law(Y)=\nu$, ...
- 109k
3
votes
1
answer
304
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Question abouth Skorokhod representation of random variables
It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e.
$$\rho(\mu,\nu)<\varepsilon,$$
then there exist two random ...
- 128k
2
votes
1
answer
145
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Calculate correlation values of an ensemble of $N\times N$ real asymmetric random matrix from Gaussian measure
I am now reading a paper by Sommers, H. J., et al. "Spectrum of large random asymmetric matrices." Physical Review Letters 60.19 (1988): 1895-1898., it claims a mathematical statement (equation (2) in ...
- 188k
3
votes
1
answer
585
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Classification of Lebesgue-Rokhlin spaces
I am currently trying to grasp some ideas on Lebesgue-Rokhlin spaces from Bogachev, "Measure Theory", vol. 2.
Such spaces are also known as standard probability spaces but the definitions are not ...
CommunityBot
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3
votes
0
answers
84
views
Stochastic equation
Let $X,Y$ be Polish spaces and $\kappa:X\times \mathcal B(Y)\to[0,1]$ be a Borel-measurable stochastic kernel on $Y$ given $X$. Under which conditions for a probability measure $\nu$ on $Y$ there ...
- 1,655
11
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1
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2k
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Do Measurable Cardinals Exist? (assuming ZFC)
In Appendix B of his Uniform Central Limit Theorems (1999), Dudley writes:
It is consistent with the usual axioms of set theory (including the axiom f choice) that there are no measurable cardinals, ...
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0
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216
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Hoeffding's lemma for unbounded r.v with bounded exponential map
Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty $ for all $\lambda \in [-c,c]$.
Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: $$E[...
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0
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81
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Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?
According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that :
$$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$
where:
$X$ is separable real Banach space.
$\...
- 121
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4
answers
1k
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Reference for a strong intermediate value theorem for measures
Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...
- 10.5k
32
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1
answer
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Do invariant measures maximize the integral?
Update: The negative answer to the following question has been provided by Matthew Daws, who won, but also rejected, the bounty of 100 euro that I set over the question.
Let $\mathcal M(\mathbb Z)$ ...
- 63.9k
5
votes
2
answers
898
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Density of Gaussian measures on Banach spaces
I am trying to get my head around this question and was reading (1) which states the same a little bit more general:
Let $X$ be a separable Banach space and $X^*$ the dual space. The mean
value $...
CommunityBot
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2
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0
answers
225
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Stopping time sigma-fields
Let $(F_n)$ be a discrete Filtration and $S_n,S$ (not necessarily finite) stopping times with $S_n\uparrow S$ (increasing convergence).
Is it true that the associated sigma-fields satisfy $F_{S_n}\...
- 29.8k
8
votes
1
answer
360
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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 $\mathcal{P}(X, \tau)$?
- 34.7k
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1
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Does Borel's proof for existence of normal numbers make an essential use of axiom of choice?
A normal number is a real number whose infinite sequence of digits in every base $b$ is distributed uniformly in the sense that each of the $b$ digit values has the same natural density $\frac{1}{b}$, ...
3
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0
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237
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Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...
- 10.5k
-1
votes
2
answers
440
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$\langle X\rangle_t = t$
Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^d$ and $X_t = |B_t|$. What is the easiest way to see that$$\langle X\rangle_t = t?$$I need this result for a simulation I am running...
- 5,005
2
votes
0
answers
121
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Coupling Marginals of Distributions on the Sphere
Given a distribution $P_X$ on $\mathbb{R}$, when does there exist a coupling (i.e. joint distribution) $P_{X^n}$ of $X_1,...,X_n$, each distributed according to $P_X$, such that $\sum X_i^2 = n$ ...
- 31
2
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0
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60
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A canonical example of the non-existence of predictive probability distribution
Section 3 of Fortini et al. (2000) states that
Given $(X^\infty, \mathcal X^\infty,P)$, a predictive probability distribution of $x_n$ given $(x_1, \dots, x_{n-1})$ with respect to $P$ need not ...
- 39.9k
11
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2
answers
3k
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Good examples of random variables whose image is not a measurable set?
Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"?
I am teaching Doob's lemma (for two real-valued ...
- 41.1k
2
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1
answer
446
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Is the following "section-wise" defined function measurable in the product space?
I asked this question in mathstackexchange a couple of days ago. Almost right after posing it a partial (affirmative) answer came to my mind in the following form
Proposition: Assume that $(X,\...
CommunityBot
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1
vote
0
answers
146
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Measurable selections of a finite familiy of measures
EDIT. I'm adding a missing hypothesis and a really TL;DR version of the core problem. Warning: This short statement is the strongest form of what I want, hence not as plausible as the original form.
...
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