All Questions
Tagged with pr.probability measure-theory
823 questions
0
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102
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Law of a step function and its generalization to two dimensions on an appropriate spaces
Let's consider two discontinuous functions defined on $D$ and $D \times [0,T]$, respectively:
A step function: $u_1(x)=\begin{cases}
u_{L}, x<c_1, \\[2ex]
u_{R}, x>c_1,
\end{cases}$
A "...
- 128k
1
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0
answers
169
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A question about Stroock's notes on the Weyl lemma
On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
- 1,461
14
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4
answers
4k
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What are two independent, uniformly distributed random variables on the unit interval?
I have been dabbling in learning basic things about probability theory and (of course) being of the school of abstract nonsense I have tried to understand things in its language. I apologize if this ...
26
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3
answers
11k
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L1 distance between gaussian measures
L1 distance between gaussian measures: Definition
Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...
- 6,853
1
vote
1
answer
84
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Disintegration associative
Is the disintegration of two borelian probabilities measures is associative ? It means if $\mu = \mu_{y}^{1} \oplus h_{\#}^{1}\mu$ and $ h_{\#}^{1}\mu = \mu_{y}^{2} \oplus h_{\#}^{2} h_{\#}^{1}\mu$. ...
- 179
2
votes
1
answer
238
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Thermodynamic limit and Gaussian measures
Let $\Lambda \subset \mathbb{Z}^{d}$ be finite and fixed and consider $\mathbb{R}^{|\Lambda|}$ be the vector space of all sequences $\varphi = (\varphi_{x})_{x\in \Lambda}$. We equip $\mathbb{R}^{|\...
2
votes
1
answer
346
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Relate the solid angle and surface measure of a surface
Let $M$ be a 2-dimensional embedded $C^1$-submanifold of $\mathbb R^3$ with a global chart$^1$ $(U,\phi)$. If $u\in U$ and $x=\phi^{-1}(u)$, let $\nu_M(x)$ denote the unique unit normal vector of $M$ ...
CommunityBot
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6
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2
answers
1k
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Is the separability of the space needed in the proof of the Prohorov's theorem?
The Section 5 of the book:
Billingsley, P., Convergence of Probability Measures, 1999,
studies Prohorov's theorem. A short reminder is given below.
Let $\Pi$ be a family of probability measures on ...
- 29.8k
0
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1
answer
284
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Explicit examples of (probability) measures on $\prod \mathbb{R}$
Let $\prod_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some ...
- 5,405
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83
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Embedding random variables in infinite-dimensional spaces
Let $H$ be a reproducing kernel Hilbert space of functions $f:E\to F$ with kernel $k$. A point in $E$ may be embedded into $H$ via the canonical embedding $x\mapsto k(x,\cdot)$. Similarly, a random ...
- 710
1
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0
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105
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Measure on a set and its value on $\emptyset$
After my first post here, I have one more doubt which is bothering me. It concerns Minlos's book Introduction to mathematical statistical physics again. To fix the notation, we have $\Lambda \subset \...
- 1,305
1
vote
1
answer
184
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Measure, volume and cardinality on Minlos' book on statistical physics
The following content was based on Minlos' book on statistical physics. Let $\Lambda \subset \mathbb{R}^{d}$ be fixed (Minlos takes $d=3$ but I think the ideas follow without change to $d \ge 1$). We ...
- 128k
4
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2
answers
267
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Grand-canonical Gibbs measure for continuous systems
Let's consider a bounded (maybe compact) set $\Lambda \subset \mathbb{R}^{d}$ with particles interacting on it. Suppose, for each $N \in \mathbb{N}$, $U_{N}: (\mathbb{R}^{d})^{N} \to \mathbb{R}\cup \{+...
- 54.2k
3
votes
2
answers
421
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Two independent function when considered as random variable over $([0,1],\mathrm{Lebesgue},\mathcal{B}_{[0,1]})$
Does there exists two non-constant continuous functions $f,g:[0,1]\rightarrow \mathbb{R}$ so that they are in independent (in probability sense) when viewed as random variables over the measure space $...
- 1,989
1
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1
answer
448
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Law of large numbers for random Dirac measures
Suppose $\{X_1,...X_n\}:\Omega \to \mathbb{R}^p$ be i.i.d. random vectors with common probability law/measure $p$, i.e. $Prob(X_i^{-1}(E))=p(E) \forall E \subset \mathbb{R}^p $ Borel measurable.
...
- 128k
7
votes
1
answer
1k
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Properties of convolutions
Consider the function
$$f_{n}(x)=e^{-x^2}x^n.$$
and the function
$$h_p(x):=e^{-\vert x \vert^p}.$$
My goal is to analyze
$$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*...
CommunityBot
- 1
0
votes
0
answers
221
views
Distance between two sample quantiles
Let $X_1,\dots X_n$ be i.i.d. samples from an unknown distribution. We know the distribution has uniformly bounded probability density function $f(x)$. Let $1>\tau_1>\tau_2>0$ be two quantile ...
- 323
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0
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96
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If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?
Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
- 167
2
votes
2
answers
294
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Imprecise Definition of a $\sigma$-algebra
I'm reading some works on the hierarchical model in statistical mechanics and I came across an strange definition, which I need to clarify. Consider a finite set $\Lambda \subset \mathbb{Z}^{d}$. The ...
- 3,535
1
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1
answer
176
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Gaussian Property of the Renormalization Group
Let $\Lambda \subset \mathbb{Z}^{d}$ be a finite set and $\varphi = (\varphi_{x})_{x\in \Lambda} \in \mathbb{R}^{|\Lambda|}$. Let $F^{\Lambda}=F^{\Lambda}(\varphi)$ be a real-valued global function, ...
5
votes
2
answers
642
views
Is the Hausdorff metric on sub-$\sigma$-fields separable?
Let $(X,\mu,\mathcal{F})$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras) of $\mathcal{F}$ ...
- 3,312
0
votes
1
answer
190
views
Visualization of the disintegration theorem [closed]
Where can I find a picture that gives a visualization of the disintegration theorem?
If such reference does not exist, what would a nice visualization of this fundamental result look like?
- 17k
6
votes
4
answers
659
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Absolutely continuous coupling of probability measures
I have a Borel probability measure $\pi$ on $\mathbb{R}^{n+1}$ such that $\pi_1=\mu_1, \ldots, \pi_{n+1}=\mu_{n+1}$ for some fixed Borel probability measures $\mu_1, \ldots, \mu_{n+1}$ (where each $\...
- 17k
17
votes
2
answers
1k
views
Is measure preserving function almost surjective?
Let $F:[0,1]\to[0,1]$ be a Lebesgue measure preserving function. Is $F$ almost surjective, i.e., the image of $F$ has interior measure one?
This question is motivated by the following observation. If ...
- 109k
0
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1
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218
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Absolute continuity of limiting measures
Let $A_n$, $B_n$ for $n \in \mathbb N$ be finte subsets of compact set $X$ in $\mathbb C$ such that
$A_n \subset B_n$.
Let $\delta_{A_n}:= \frac{1}{|A_n|} \displaystyle\sum_{x\in A_n} \delta_x$ and $\...
- 128k
1
vote
1
answer
206
views
Almost identical $\sigma$-algebras and measurability
Let $(X,\mathscr X,\mathbb P)$ be a probability space, $(Y,\mathscr Y)$ a measurable space, and $h:X\times Y\to\mathbb R$ a real-valued function measurable with respect to the product $\sigma$-algebra ...
- 175
1
vote
1
answer
120
views
Are there well-established notions of convergence of measures that take into account differentiable structure?
All the notions of convergence of measures that I know of are either in the purely measure-theoretic category (e.g. strong convergence, total variation), or in the topological category (e.g. weak ...
- 128k
1
vote
1
answer
271
views
Measurable selection
The comments in the post
Almost sure identity
has led me to the following question:
Suppose that $(\Omega,\mathcal{F},P)$ is a probability space.
Denote by $\mathcal{B}(\mathbb R)$ the Borel ...
- 1,769
2
votes
2
answers
451
views
If $0 \le \mu(A) < p < 1$, when is it true that there exists a measurable $B \supseteq A$ such that $\mu(B)=p$?
Let $(X,\mu)$ be a probability measure space and $A$ be a measurable subset of $X$ such that $0 \le \mu(A) < p < 1$.
Question
When is it true that there exists a measurable $B \subseteq X$ ...
- 109k
1
vote
1
answer
860
views
Right continuous filtration
In optimal control theory, we often need a filtration do be right continuous. Consider a filtered probability space $(\Omega, \mathcal F, \mathbb P)$ equipped with a right continuous filtration $\...
- 17.2k
1
vote
0
answers
106
views
Show a Poincaré inequality for a Markov kernel and minimize the Poincaré constant
Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$ (see definitions ...
- 167
1
vote
0
answers
56
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Minimizing the rate of geometric ergodicity of a Metropolis-Hastings kernel depending on a parameter
Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$.
I want to ...
- 167
2
votes
1
answer
99
views
Does bounded integral over sequence of subsets of $X$ whose union is $X$ imply bounded integral over X?
I came across the following problem while doing a piece of research on automata theory.
Suppose we have a probability space $(\Omega, \mathcal{F}, \mu)$, where $\Omega$ is a set, $\mathcal{F}$ is a $\...
- 17.2k
9
votes
1
answer
4k
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What are some characterizations of the strong and total variation convergence topologies on measures?
I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.
The Wikipedia article on convergence of measures defines three kinds of convergence: ...
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2
votes
1
answer
139
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Properties of Random and Stopping Sets
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $S$ be a countable set, and we define a family of sub-$\sigma$-algebras $(F_A)_{A \subseteq S}$ such that $A \subseteq B \Rightarrow \...
- 128k
2
votes
2
answers
801
views
Weak convergence in Skorohod topology
Let $D([0,T];R^d)$ be the space of càdlàg functions endowed with the usual Skorohod topology. $X_t(\omega):=\omega(t)$ denotes the usual canonical process. Assume that a family of probability ...
- 1,989
4
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1
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203
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"Inner Regularity" of probability measure on totally ordered sets
I am looking for an example:
Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space, where $\Omega\subseteq \mathbb{R}^n$ (for some $n\in \mathbb{N}$) is a totally ordered set with ...
- 32.5k
3
votes
1
answer
1k
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When are events in tail $\sigma$-algebra the limsup of some sequence of events?
Consider a sequence of $\sigma$-algebras $\mathcal{F}_1,\mathcal{F}_2,\dots$. Is it true that for any event $B$ in the tail $\sigma$-algebra $\mathcal{F_{\text{Tail}}}$, it can be expressed as the $\...
- 24.8k
5
votes
1
answer
652
views
Proof of Pinelis (1992) - Banach space inequalities
I am reading Pinelis "An approach to inequalities for the distributions of infinite -dimensional martingales" and cannot follow his proof of Theorem 3:
Let $(f_n)$ be a martingale in a separable ...
- 128k
5
votes
2
answers
791
views
What is this disintegration-like theorem?
This is cross-posted at MSE.
I'm looking for a reference for the following result. It seems like it must be known, or follow quickly from something known, but I have not been able to find it in any ...
- 1,769
-1
votes
1
answer
83
views
Convergence in mean and convergence in distribution
Suppose a sequence of random variables $X_n$ convergence in distribution to $X$, and $Y_n$ convergence in pth-mean (any $p\geq 1$) to $Y$. Moreover, there exist constants $c_0,c_1$ such that
$$
0< ...
- 3,937
1
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0
answers
63
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Approximation of measured-valued function by continuous functions
For each $x\in R^d$, let $\nu(x,dz)$ be a L\'evy measure, i.e.,
$$
\int_{R^d}(|z|^2\wedge1)\nu(x,dz)<\infty.
$$
Let $\mu$ be a probability measure on $R^d$ such that
$$
\int_{R^d}\int_{R^d}(|z|^2\...
- 411
1
vote
1
answer
120
views
Conditional distribution/independence
Suppose $X,Y$ and $Z$ are random elements on $(\Omega,\mathcal{A},\mathit{P})$ taking values in the Borel spaces $U,V$ and $V$ respectively. Moreover, let $\mathcal{F}\subset \mathcal{A}$ be a $\sigma-...
- 128k
1
vote
1
answer
178
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Predictable Projection of a Stopped Process (Typo in Jacod & Shiryaev?)
Given a filtered probability space $( \Omega, \mathcal{F}, (\mathcal{F}_t)_t, \mathbb{P})$ and an $\mathcal{F} \otimes \mathcal{B}(\mathbb{R}_+)$-measurable bounded process $X: \Omega \times \mathbb{R}...
- 1,989
0
votes
0
answers
75
views
Optimizer of a semi-discrete optimal transport problem
Provided two probability distributions $\mu(dx)=\rho(x)dx$ and $\nu(dx)=\sum_{i=1}^n p_i\delta_{y_i}(dx)$ that are supported on some measurable set $\Omega\subset\mathbb R^d$, we consider the semi-...
CommunityBot
- 1
1
vote
0
answers
91
views
Probability space with countable subset such that every subset of positive measure meets the subset
Let $(X, \mathcal F, P)$ be a probability space.
Question
What kind of condition is this: there exists a sequence $(a_n)_n \subseteq X$ such that
$\forall$ measurable $A \subseteq X$, $P(A) >...
- 63.9k
3
votes
1
answer
372
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Attractors in random dynamics
Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...
- 63.9k
3
votes
2
answers
2k
views
Empirical estimator for total variation distance between two product distributions
Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...
- 6,541
2
votes
0
answers
56
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What is the Wiener measure of the set of curves with given Hölder constant on a Riemannian manifold?
Let $M$ be a connected Riemannian manifold and $x_0 \in M$. For $0 < \alpha < \frac 1 2$, let
$$H = \{ c : [0,1] \to M \mid c(0) = x_0 \text{ and } \exists C>0 \text { s.t. } d(c(s), c(t)) \...
CommunityBot
- 1
1
vote
1
answer
173
views
Can we show that $\mathbb R^{\mathbb N}\ni x\mapsto\bigotimes_{n\in\mathbb N}\mathcal N_{x,\:\sigma^2}$ is a Markov kernel?
Let $\sigma>0$ and $\mathcal N_{x,\:\sigma^2}$ denote the normal distribution with mean $x\in\mathbb R$ and variance $\sigma^2$. From the Ionescu-Tulcea theorem, we know that $$\kappa(x,\;\cdot\;):=...
- 167