All Questions
87 questions
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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
0
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1
answer
133
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Product of sets with the Radon-Nikodym Property (RNP)
I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP.
Does the above result ...
- 59
7
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1
answer
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Reference request: norm topology vs. probabilist's weak topology on measures
Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...
- 3,486
4
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2
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667
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Convergence (topology) for $\sigma$-finite measures
I'm having much trouble finding literature that addresses the questions which I write below. I was wondering if someone could help me out to understand better, either by providing references or by ...
- 29.8k
2
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1
answer
235
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Kolmogoroff condition for truncated random variables
Question summary. Does the Kolmogoroff condition $\sum_{n=1}^\infty\frac{\mathbb V Y_n}{n^2} < \infty$ hold for truncated random variables $Y_n := X_n \cdot 1_{\{X_n \le n\}}$ (see below for a more ...
- 128k
1
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0
answers
162
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Does the law of a Feller process depend continuously on the initial condition?
Let $E$ be a locally compact and separable metric space, and suppose $X$ is a Feller process with transition function $P_t$. To be precise, let $C_0$ denote the space of continuous functions vanishing ...
- 11
1
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0
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192
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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 ...
- 63.9k
0
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0
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424
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Bounding the total variation distance between two measures from a given set
I have a distance on the space of probability measures on $[0,2]$. It is defined as such for two probability measures $\mu_1$ and $\mu_2$ :
$d_p(\mu_1,\mu_2) := \sum_{k=0}^p ( \mathbb{E}[X_1 ^k]- \...
- 31
1
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0
answers
58
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Extension of a result about measurable, additive functionals
Let $W$ be a set, and let $v$ be a finitely additive probability measure on $2^W$.
Equip $2^W$ with the Borel sigma-algebra $\mathcal{B}$ generated by the sub-basic sets of the form $\{a: w \in a\}$ ...
- 869
5
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1
answer
386
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Lower semi-continuity of the Hellinger-Fisher-Rao distance
I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance
$$
H^2(\rho,\mu)=\int_{\Omega}\left|\sqrt{\frac{d\rho}{d\lambda}}-\sqrt{\frac{d\mu}{d\...
- 1,027
6
votes
1
answer
196
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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, ...
- 3,574
1
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0
answers
120
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Existence of Time-Reversed Markov Kernels
Suppose I have a probability measure $\pi$ and a Markov kernel $q$ which leaves $\pi$ invariant, in the sense that
\begin{align}
\int_x \pi(dx) q(x \to dy) = \pi(dy)
\end{align}
Then, a (the) time-...
- 801
2
votes
1
answer
133
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Optimal-score partitions
The question about throwing darts asked on the MathOverflow page Sacred Geometry of Chance was not well received, apparently because of "[t]oo much noise around the actual math", as stated in a well-...
- 128k
4
votes
1
answer
188
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Absolute continuity of measures - reference sought
For two measures $\mu, \nu$ on the same space say that $\mu$ is absolutely continuous with respect to $\nu$ ($\mu \ll \nu$) whenever $\nu(A)=0$ implies that $\mu(A)=0$ too.
Let $(\Omega, \mathsf P$) ...
- 128k
12
votes
3
answers
870
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Measure theory in nuclear spaces
Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
6
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3
answers
938
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Uniformly distributed sequence in $\mathbb{R}$
We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and
$$\lim_{N \to \infty} \...
- 4,713
3
votes
1
answer
732
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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
700
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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
20
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1
answer
2k
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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
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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
3
votes
1
answer
940
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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
4
votes
1
answer
1k
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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
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
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
19
votes
3
answers
3k
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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
6
votes
1
answer
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
- 1
18
votes
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
3
votes
0
answers
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
8
votes
1
answer
726
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continuity of the Boltzmann entropy in the Wasserstein metric
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
- 23.2k
1
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0
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417
views
Defining density of a random function using Radon-Nikodym Theorem
Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$.
Let $X$ be random function defined ...
- 213
7
votes
1
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453
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Do the terms of an iid sequence whose law has infinite expected value necessarily exceed the partial sums of the sequence infinitely often?
Let $\mu$ be a probability measure on $(0,\infty)$, and let $(\mathbf X_n)_1^\infty$ be a sequence of independent $\mu$-distributed random variables. Fix $\kappa > 0$, and consider
A) $\int x \; d\...
- 1,136
3
votes
1
answer
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 ...
- 591
5
votes
1
answer
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
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 ...
8
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0
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729
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Density of countably additive measure in the set of all finitely additive measures.
Let $S$ be a countable discrete set, the following two results are quite easy to prove:
Every countably additive probability measure $\mu$ on $S$ commutes (in Fubini's sense) with every finitely ...
- 6,034
12
votes
2
answers
3k
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Does there exist an event independent of a given sigma-algebra?
The following question came up in a discussion with my advisor:
Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-...
- 17.1k
10
votes
2
answers
1k
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Continuity of the mutual information
The mutual information $I(\mathfrak A_1;\mathfrak A_2)$ of two complete $\sigma$-algebras $\mathfrak A_1$ and $\mathfrak A_2$ in a Lebesgue probability space $(X,m)$ is the integral of the logarithm ...
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