All Questions
Tagged with pr.probability measure-theory
823 questions
4
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PDE-Based Triangle Inequality for Optimal Transportation
Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and $\...
CommunityBot
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4
votes
1
answer
740
views
Integral wrt probability measure
Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P_\theta$ is a probability measure on $(\cal X, \cal A)$. ...
CommunityBot
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3
votes
1
answer
226
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conditional expectation under convex combinaison of probability measures
Let $(\Omega,\mathcal{F})$ denote some measurable space. Let $P_1$ and $P_2$ denote respectively two probability measures. Now let $\mathcal{G}$ be some sub sigma-algebra of $\mathcal{F}$. Given a ...
- 3,958
2
votes
1
answer
91
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Measures of disjoint unions and complements of a collection of sets
Let $\mu$ be a probability measure. Let $\mathcal A$ be a collection of measurable sets and $D(\mathcal A)$ be the minimal $\lambda$-system (Dynkin system) containing $\mathcal A$.
Is $\mu(D)$ for $...
- 24.8k
3
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0
answers
164
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Existence of a conditional distribution
Let $X$ and $Y$ be standard Borel spaces and let $J$ be an analytic subset of $X\times \mathcal P(Y)$ where $\mathcal P(\Omega)$ is a set of probability measures on a Borel space $\Omega$ endowed ...
- 1,655
4
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0
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309
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Conditional expectation with respect to random closed sets
Short question
If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked ...
- 6,287
4
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1
answer
1k
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General version of Skorokhod representation of random variables
Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the ...
- 487
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4
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3k
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Convergence of probability measure and the *-weak convergence ?
Given a Polish space $X$, I note $C_b(X)$ the set of the continuous bounded functions with the norm of the uniform convergence, and $(C_b(X))^\star$ its topological dual with the $*-$weak convergence $...
- 166
6
votes
3
answers
1k
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Is the set of the absolutely continuous functions a Borel set of the space of the continuous functions?
Does anyone knows whether the set of the absolutely continous functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the ...
- 29.8k
-1
votes
1
answer
259
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Absolute continuity of probabilities on Polish spaces and open sets. [closed]
On a polish space $\mathcal{X}$ i consider two Borel probabilities $P$ and $Q$ such that for any open set $E$ of $\mathcal{X}$ we have : $P(E) =0$ implies $Q(E)=0$. Does this imply that $Q$ is ...
CommunityBot
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5
votes
0
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597
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Skorohod theorem (weak convergence) on a discrete setting
I have a question about the application of Skorohod representation theorem. The questions arises in this paper about robust hedging in mathematical finance. It is about the very last equation on page ...
CommunityBot
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5
votes
1
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758
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Cameron-Martin theorem for non-Gaussian measures
Let $X$ be a locally convex topological linear space, and $\mathbb P$ be a probability measure on $X$. Denote the mean vector $m \in X$ and covariance operator $k : X^* \to X$. Let $\tau_u : X \to X$ ...
- 29.8k
4
votes
1
answer
580
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Density of linear functionals in $L^2$
Let $X$ be a locally convex topological linear space, and let $\mathbb P$ be a probability measure on $X$. Suppose that $\operatorname{var}(\varphi) < \infty$ for all continuous linear functionals $...
- 29.8k
1
vote
0
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146
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approximation of probability distribution
I have a question: Let $\mu$ be a probability distribution defined on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ satisfying
$$\int_{\mathbb{R}}|x|d\mu<+\infty$$
Set
$$A_n=\Big\{\frac{i}{n}:~ i\in\...
- 293
2
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0
answers
136
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equivalence of topologies defined on $M_1$(a subspace of bounded measures on $\mathbb{R}$)
Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded elements. Define for $\phi(x):=1+|x|$,
$$
\...
- 3,958
0
votes
1
answer
137
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Existence of a map connecting two marginals of a product measure
Let $X$ and $\bar X$ be two standard Borel spaces, and let $A\subseteq X\times\bar X$ be an analytic subset of the product space. Let $P$ be any probability measure such that $P(A) = 1$, and denote by ...
- 29.8k
3
votes
1
answer
290
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A question about stochastic kernels and invariant measures
Suppose that $E$ is a metric space, let $\mathcal{B}_E$ denote the set of its Borel subsets and suppose that $\mu$ is a probability measure on $(E,\mathcal{B}_E)$. In addition, suppose that $p:E\times ...
- 3,958
2
votes
1
answer
192
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Particular neighborhoods of analytical sets
Let $X$ be a standard Borel space: a topological space isomorphic to a Borel subset of a complete separable metric space. Denote by $\mathcal P(X)$ the set of all Borel probability measures over $X$ ...
CommunityBot
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3
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1
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2k
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From Lebesgue Integral to Stieltjes Integral, and integration by parts
Let $X$ be a real random variable with c.d.f function $F$.
Let $g$ be an increasing measurable real function and assume that $\mathbb{E}\left[g(X)\right]$ exists (and is finite).
What additional ...
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11
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2
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1k
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Is there a measure / probability theory in a topos of "generalized measure spaces"?
Consider the category with the standard Lebesgue measure space $\Omega$ as its only object and measure type preserving nonincreasing (equivalence classes of) maps as morphisms.
Question: is there an ...
CommunityBot
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5
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1
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dual space of the subspace of the space of probability measures [closed]
I have a question which maybe so naive but I want to know the result about it.
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Then by some materiau such as ...
- 16.4k
2
votes
1
answer
245
views
Probability measures on $L^p$
Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...
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2
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1
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414
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When is a space of probability measures not perfectly normal?
I am looking for examples of pairs ($(\Omega,\Sigma)$, ($\mathcal P(\Omega)$, $\tau$)), where $(\Omega,\Sigma)$ is a measurable space and ($\mathcal P(\Omega)$, $\tau$) is a space of probability ...
CommunityBot
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3
votes
3
answers
924
views
A non-trivial probability measure on $2^{\mathbb R}$
Consider the measurable space $2^{\mathbb R}$, equipped with the tensor-product $\sigma$-algebra. Famously, this space has a measurable structure which is not generated by a topology (see this answer)....
CommunityBot
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-2
votes
1
answer
347
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Forms of multivariate CLT [closed]
I am looking for a good reference for differnt kinds of multivariate central limit theorems. I was wondering how far the i.i.d. condition of the standard multivariate clt can be relaxed, as in can the ...
- 7,514
16
votes
4
answers
1k
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Continuity on a measure one set versus measure one set of points of continuity
In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?
Now more carefully, with some notation: Suppose $(X, d_X)$ ...
- 44.4k
17
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5
answers
3k
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Conditional probabilities are measurable functions - when are they continuous?
Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
- 311
2
votes
1
answer
557
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Is this a closed set?
Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of ...
- 60.5k
3
votes
1
answer
651
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What conditions on a probability distribution defined by long-time averaging do I need to satisfy a central limit theorem?
For integer $n$, $1 \le n \le N$, consider the random variables
$X_n = \cos[t \omega_n]$
For any fixed $N$, we can take the mean
$Y_N = \frac{1}{N} \sum_{n=1}^N X_n$
and define a (cumulative) ...
- 19.3k
3
votes
1
answer
354
views
Determining the asymptotic behavior of random matrices with vanishing ratio dimensions
Consider an $N\times K$ random matrix $X$ (defined on a probability space $(Ω,F,μ)$) with i.i.d. entries having zero mean and variance $1/K$.
There are a lot of results regarding the asymptotic ...
- 7,514
5
votes
1
answer
585
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Existence of a probability measure with "confined" zero measure sets
Hi, I am struggling with the following question that is tangentially arising from a paper I'm working on. It is not at all essential for the revision but it would be nice to know if there is a ...
- 60.5k
8
votes
0
answers
212
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Qualitative weakenings of probabilistic independence
In probability theory, independence of random variables is characterised by
$$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$
where $P_{(X,Y)}$ is the joint probability ...
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2
votes
1
answer
3k
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Empirical estimator fot the total variation distance on a finite space
I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$):
$$...
- 7,514
6
votes
2
answers
318
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Borel kernel over an analytic set implies existence of a Borel map
Let $X$ and $Y$ be standard Borel spaces, and let $A\subseteq X\times Y$ be an analytic set with a full projection on $X$: that is $\pi_X(A) = X$. Suppose that there exists a Borel-measurable kernel $\...
- 17.1k
23
votes
1
answer
3k
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Bochner integral of stochastic process = path by path Lebesgue integral?
After some helpful comments, I realized that I had to repost this question in a more systematic way.
On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...
- 17.1k
17
votes
2
answers
860
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A moment problem on $[0,1]$ in which infinitely many moments are equal
Suppose $\mu$ and $\nu$ are two probability measures on $[0,1]$. Let their $n$-th moments be denoted by $\mu_n$ and $\nu_n$, respectively, for $n \in \mathbb{N}$.
If we know that $\mu_n=\nu_n$ for ...
CommunityBot
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1
vote
1
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483
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A set of positive-measure not being a countable union of cylinder sets and zero-measure sets?
Let $(A^\mathbb{N}, \mathcal{B}(A^\mathbb{N}), \mu)$ be a measure space, where $A^\mathbb{N}$ is a set of one-sided sequences over a finite alphabet $A \subset \mathbb{N}$, $\mathcal{B}(A^\mathbb{N})$ ...
- 6,206
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2
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649
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What's the standard name for sets of a given size with maximal probability (or a given probability and minimal size)?
The definition I'm going to give isn't quite the concept I really want, but it's a good approximation. I don't want to make the definition too technical and specific because if there's a standard name ...
- 833
5
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1
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492
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Coupling of non-probability/sub-probability measures
A coupling of two probability measures $P,\tilde P$ on a Borel space $X$ is any probability measure on $X^2$ whose one-dimensional marginals are $P$ and $\tilde P$. In particular, for any such ...
- 12.7k
3
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1
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275
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Conditional probabilities for all pairs of subsets in $\mathbb R^2$?
Parikh and Parnes showed that one can define a isometry-invariant finitely-additive conditional probability for all pairs of subsets of the interval. Can one extend this result to bounded subsets of $...
- 2,555
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2
answers
1k
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Why do we want maps to be measurable (in countably-additive setting)
When I have to explain things that I am doing to people who did not do (or even did not learn) measure-theoretical probability, I think of getting a question in the title, and I am not sure I have ...
- 1,655
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0
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416
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When does a proper Zariski closed set have measure zero with respect to a conditional measure?
Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure.
Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
- 6,210
3
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1
answer
263
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Extending Tarski's Theorem on invariant measures
Tarski's Theorem says that if $G$ acts on $X$ and $E$ is a non-$G$-paradoxical subset of $X$, then there is a finitely additive $G$-invariant measure $\mu:2^X\to[0,\infty]$ with $\mu(E)=1$.
I am ...
- 352
5
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1
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469
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Universally measurable map coincides a.e. with a Borel map
Let $X$ be a standard Borel space: that is, a topological space equivalent to a Borel subset of $\Bbb R$. It is known that for any probability measure $p$ on $X$ and any universally measurable set $A\...
3
votes
1
answer
143
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Maps that are a.e. equal have almost the same graphs
Let $X$ and $Y$ be two measurable spaces, and let $p$ be a probability measure on $X\times Y$. Denote by $p_X$ the marginal of $p$ on $X$, that is an image of $p$ under projection on $X$. Consider two ...
- 60.5k
4
votes
3
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3k
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What is the name for a non-normalized distribution?
For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work ...
CommunityBot
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6
votes
2
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720
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Local concentration of measure on Erdos-Rényi graph
Let $G_n=(V_n,E_n)$ be an Erdos-Rényi random graph, precisely the vertex set is $V_n=(1,\dots,n)$ and the edge set is $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \epsilon_{ij}=1)$ where $(\epsilon_{ij})_{ij}$ ...
- 7,514
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0
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1k
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Measure induced by function
It is known that an n-increasing, left-continuous function $f$, on $[0,\infty)^n$ induces a unique positive measure $\mu$ on $[0,\infty)^n$. Say if $f$ was 3-increasing on $[0,\infty)^3$ but also 2-...
- 8,271
6
votes
1
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798
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Prohorov's theorem for random elements of Hilbert space: weak convergence
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(E,\mathcal{E})$ be a separable Hilbert space ($E$) with Borel $\sigma$-algebra $\mathcal{E}$. For concreteness let us set $E=L^{2}[a,b]$ ...
- 23.2k
8
votes
1
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1k
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Conditional law as a random measure and convergence of random measures
I'm looking for a reference book or article for the following two facts. In both statements, a Polish space $E$ and an ambient probability space $(\Omega, {\cal A}, \Pr)$ are given, and I consider ...
