All Questions
Tagged with gn.general-topology pr.probability
24
questions with no upvoted or accepted answers
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 ...
- 899
7
votes
0
answers
277
views
Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,322
6
votes
0
answers
183
views
Pettis Integrability and Laws of Large Numbers
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...
- 8,322
6
votes
0
answers
679
views
What is the structure of a space of $\sigma$-algebras?
Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
- 8,322
4
votes
0
answers
94
views
Is the range of a probability-valued random variable with the variation topology (almost) separable?
Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
- 12.3k
3
votes
0
answers
117
views
Eigenvalues of random matrices are measurable functions
I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable.
If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
- 31
3
votes
1
answer
147
views
Existence of disintegrations for improper priors on locally-compact groups
In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...
- 8,322
3
votes
0
answers
143
views
Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?
I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
- 170
3
votes
1
answer
971
views
"Relative compactness of a family of probability measures" and relative compactness & sequential compactness of sets
I'm studying Billingsley's convergence of probability measures, and wondering why the definition of "Relative compactness of a family of probability measures" reasonable.
In the discussion ...
- 31
2
votes
0
answers
45
views
$\sigma$-compactness of probability measures under a refined topology
Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
- 195
2
votes
0
answers
136
views
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 $\phi(x):=1+|x|$,
$$
\...
- 1,827
2
votes
0
answers
140
views
Products for probability theory using zero sets instead of open sets
(For all of this post, at least Countable Choice is assumed to hold.)
For all Tychonoff spaces $\langle X,\mathcal{T}\hspace{.06 in}\rangle$ :
Define $\mathbf{Z}(\langle X,\mathcal{T}\hspace{.06 in}\...
user5810
1
vote
0
answers
81
views
Reference request: rates of weak convergence of Polish space-valued random variables
Let $(E,\mathscr{E})$ be a Polish space $E$ together with its Borel $\sigma$-algebra $\mathscr{E}$. Let $(\Omega, \mathscr{F},P)$ be a probability space and let $(X^{(n)})_{n \in \mathbb{N}}$ be a ...
- 131
1
vote
0
answers
77
views
Density of Lipschitz functions in Bochner space with bounded support
Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-...
- 21
1
vote
0
answers
50
views
A local base for space of probability measures with Prohorov metric
Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
- 111
1
vote
0
answers
142
views
Clarification about the ϵ -net argument
I have been reading the paper Do GANs learn the distribution? Some theory and empirics.
In Corollary D.1, they reference the paper Generalization and Equilibrium in Generative Adversarial Nets which ...
- 19
1
vote
0
answers
66
views
Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$
Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
- 19
1
vote
0
answers
77
views
Random variables with values in binary operations or in topologies of a certain set $X$
I wonder if the following situations have already been considered by mathematicians :
Random variables with values in a set of binary operations endowed
with a certain topology (or just with a $\...
- 129
1
vote
0
answers
83
views
The role of absolute continuity in stochastic ordering defined over sets of probability distributions
This question is about a claim given in this paper (page 261, the remark), but without any proof.
It simply says that if two sets of probability distributions, $\mathscr{P}_0$ and $\mathscr{P}_1$ (...
- 349
1
vote
0
answers
33
views
Defining connectivity between K points on a periodic domain in terms of proximity
THE SITUATION:
Begin by taking a periodic strip of length 2*Pi. Then use a uniform distribution to place K points (x1,…, xk) on the strip by assigning each of them a randomly sampled number. Then ...
- 11
1
vote
0
answers
256
views
Generating the sigma algebras on the set of probability measures
I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
- 11
0
votes
0
answers
116
views
Uncountable collections of sets with positive measures
Let $X$ be a compact metric space and let $T: X \rightarrow X$ be continuous. Let $\mu$ be a $T$-invariant Borel probability measure (which we can always find by the Krylov-Bogoliubov theorem).
Let $(...
0
votes
0
answers
58
views
Specific property of borelian sigma-algebras
Let X be a set and S a sigma-algebra on X.
Let us name borelian sigma-algebra on X a sigma-algebra that is generated by a topology T on X. Given that it is possible for a set X that some sigma-...
- 2,591
0
votes
0
answers
154
views
question about the tightness of probability measures for a general topological space
Let $(E,\mathcal{X})$ be a topological space and denote by $\mathcal{F}$ its collection of Borel subsets referred to $\mathcal{X}$. Now let $\mathcal{P}$ be the set of all probabilities on $(E,\...
- 1,827