All Questions
Tagged with pr.probability measure-theory
219 questions with no upvoted or accepted answers
3
votes
0
answers
243
views
Parametric distances on product spaces of measures
Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you.
Let $X$ be a topological ...
- 6,853
3
votes
0
answers
126
views
Other than Brownian motion, when else is it possible to define "normalized weighted infinite dimensional Lebesgue measure"?
In this article Sourav Chatterjee poses the question, how do we define the measure:
$$d\mu(A)=\frac{1}{Z}\exp\left(-\frac{1}{4g^2}S_{YM}(A)\right)dA$$
The $Z$ here is an infinite normalizing ...
user69208
3
votes
0
answers
509
views
sufficient condition for the continuity of conditional probability wrt the conditioning variable
Given a regular conditional probability $P(X\in B | T(X) = t)$, where $T$ is a continuous mapping from $\mathcal{X}$ (on which $X$ is defined) to $\mathcal{T}$. Do we know any sufficient condition for ...
- 319
3
votes
0
answers
428
views
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
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
3
votes
0
answers
237
views
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
3
votes
0
answers
119
views
Reasoning about dependent and independent quantities by "degrees of freedom"
In his classic textbook Foundations of the Theory of Probability Kolmogorov defines Independence a little bit differenent then it is usually done today. He denotes a probability space by $(E, \mathcal ...
- 798
3
votes
0
answers
112
views
McDiarmid's inequality on normed spaces
McDiarmid's inequality says if a function $f: \mathcal{X}^n\to\mathbb{R}$ has the property that
$$
\sup_{x_1,\dotsc,x_n,x'_i\in\mathcal{X}} |f(x_1,\dotsc,x_n)-f(x_1,\dotsc,x_{i-1},x'_i,x_{i+1},\dotsc,...
- 619
3
votes
0
answers
465
views
How to define Product of Conditional Measures?
I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below.
If $(X,\Sigma)$ is a measurable space, then the function $\mu : \Sigma\...
- 191
3
votes
0
answers
910
views
Girsanov theorem with Geometric Brownian Motion
I am not a student in mathematics, but I am trying to use the following Theorem 8.6.6 (Girsanov theorem II) of Oksendal's SDE with geometric Brownian motion $S_{t}$ instead of the standard Brownian ...
- 41
3
votes
0
answers
206
views
Embedding probability spaces in the completion of $[0,1]^K$
Question: Can every probability space $(X,\scr F,\mu)$ be $\sigma$-embedded in the completion of the space $[0,1]^K$ (equipped with a product of Lebesgue measure) for some set $K$?
Here, $f:\scr F\to ...
- 2,555
3
votes
0
answers
164
views
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
3
votes
0
answers
133
views
What distribution(s) of delays make(s) timing attacks hardest?
$H$ is (Shannon) entropy.
In terms of the positive real number $t$, what distribution(s) $\hspace{.01 in}X$ on $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})$...
user5810
3
votes
1
answer
1k
views
Borel-Cantelli lemma for general measure spaces (those with infinite measure)
The Borel-Cantelli lemma is often stated for a probability space or spaces with finite measure.
But it seems to me that it still holds if the space $X$ is of infinite measure. I seem to be able to ...
- 31
2
votes
0
answers
92
views
Existence of ergodic subgroup invariant to a product measure
Let $X=\{0, 1\}^{\mathbb{N}}$ and $G$ be the group of permutations, each of which only permutes finitely many coordinates of $X$. Fix a sequence $(\lambda_n)_{n\in \mathbb{N}} \subseteq (0, 1]$ and ...
- 307
2
votes
0
answers
205
views
When should the empirical measure of an infinite sequence be defined?
Let $(x_n)_{n \in \mathbb{N}}$ be a (deterministic) sequence of nonnegative reals, possibly even with $x_n \in \mathbb{N}$ if you prefer. Then we'd like to define the empirical measure of such a ...
- 6,406
2
votes
0
answers
29
views
Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
- 434
2
votes
0
answers
54
views
If a probability measure is a mixture of products of its marginals, does it have finite moments?
Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$. For a linear subspace $E\subset \mathbb{R}^n$, let $\mu_E$ denote the marginal of $\mu$ on $E$. The usual orthogonal complement of $E$ is ...
- 716
2
votes
1
answer
316
views
Proof of the Dunford-Pettis theorem in the context of probability spaces
I'd like to know if there's a proof of the Dunford-Pettis theorem without using relatively advanced theorems of functional analysis such as Eberlein–Smulian Theorem. Since I'm only interested in ...
- 637
2
votes
0
answers
127
views
Measure algebra for a family of probability measures
Let $(X,B,P)$ be a probability space, $I_P$ the $\sigma$-ideal of $P$-null sets and
\begin{align}
B_P = B \ltimes I_P &= \{ A \mathbin{\triangle} N \mid A \in B, N \in I_P \}
\end{align}
the ...
- 285
2
votes
0
answers
49
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
155
views
Can a diffusion process admit an invariant measure with a non-differentiable density?
The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
- 167
2
votes
0
answers
57
views
Regularity on $\mathbb{T}^3$ of the "functional average" of a map $S : C^\infty(\mathbb{T}^3, \mathbb{R}) \to L^2(\mathbb{T}^3, \mathbb{R})$
For simplicity, let $C^\infty(\mathbb{T}^3, \mathbb{R})$ be the real Frechet space of periodic smooth functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus.
For a fixed ...
- 3,477
2
votes
0
answers
150
views
A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$
Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \...
- 657
2
votes
0
answers
92
views
A variant of disintegration theorem where the assumptions on $f$ and $g$ are exchanged
I have recently read about about disintegration theorem, i.e.,
Disintegration theorem Let
$X$ be a Polish space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X$...
- 835
2
votes
0
answers
132
views
Concentration of sample covariance for dependent data
Let $X_1, \ldots, X_T$ are sub-Gaussian random vectors in $\mathbb{R}^d$ coming from a common distribution with population covariance $\Sigma$. If they are independent, it is known that the sample ...
- 399
2
votes
0
answers
98
views
Has this "optimal constrained transport" notion of convergence of measures been named and/or studied?
Let $(X,d)$ be a compact metric space, and let $\{\mu_n\}_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures on $X$.
Fix $L \geq 1$. I will say that $\mu_n$ converges in ...
- 708
2
votes
0
answers
145
views
Are there sets in the unit cube that cannot be in the domain of any finitely-additive, isometry-invariant probability measure?
The Vitali construction implies (given choice) the existence of a set such that for any translation-invariant, countably additive probability measure on $[0,1]$, that set is nonmeasurable and has ...
- 71
2
votes
0
answers
123
views
Probability of a finite cylinder set in a free group
Let $\mathbb{F}_n$ be the free group (each elemen is in its reduced form) generated by the set $\Sigma_n = \{a_1, a_2, \cdots, a_n, a_1^{-1}, a_2^{-1}, \cdots, a_n^{-1}\}$ and let $e$ denote the ...
- 307
2
votes
0
answers
105
views
Equivalence between notions of dynamical coupling as defined by Villani in his book Optimal Transportation: Old and New
$\DeclareMathOperator\law{law}$In Villani's book he presents the following notions of dynamical couplings:
Let $(X,d)$ be a Polish space. A dynamical transference plan $\Pi$ is a probability measure ...
- 99
2
votes
0
answers
139
views
Are there any measurable spaces of functions
I am approaching this question from a probability perspective, and am hoping for some kind of framework to help understand all of this. I believe I may have even asked a similar question on here in ...
- 213
2
votes
0
answers
302
views
Simplify Kantorovich–Rubinstein duality when distributions share a common marginal
Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
- 53
2
votes
0
answers
104
views
Weak convergence rates for integral operators
Suppose $q=\sum_{i=1}^m\pi_i\delta_{x_i}$ is a discrete measure on $\mathbb{R}^n$ and let $q\ast \varphi_\epsilon$ denote the convolution of $q$ with some mollifier $\varphi_\epsilon$, so that $q\ast\...
- 75
2
votes
0
answers
142
views
Radon-Nikodým-like theorem for Radon measures
Let $(E,d)$ be a metric space, $\mu$ be a nonnegative Radon$^1$ measure on $\mathcal B(E)$ and $\nu$ be a finite (signed) Radon measure on $\mathcal B(E)$.
I'm searching for a Radon-Nikodým-like ...
- 167
2
votes
0
answers
520
views
Example of a non-reflexive Banach space and two sequences
Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$.
If $X$ is reflexive, ...
- 199
2
votes
0
answers
35
views
When does a measure-valued map admit a continuously parametrized density function?
Let $X$ and $Y$ be Polish spaces, let $\mathcal{P}(Y)$ be the space of Borel probability measures on $Y$ endowed with the smallest $\sigma$-algebra such that all functions of the form $\nu\mapsto\nu(A)...
2
votes
0
answers
61
views
Measurable extensions of probability measures
Let $X$ be a set, and let $\mathcal G \subset \mathcal F$ be $\sigma$-fields over $X$. Let $\Delta_\mathcal G$ (resp. $\Delta_\mathcal F$) be the set of probability measures on $\mathcal G$ (resp. $\...
- 869
2
votes
0
answers
261
views
Reference for Borel $\sigma$-algebra of topology of convergence in probability
I'm pretty sure I can prove the "Theorem" given further below (without very much difficulty), but it seems way too basic not to have been noticed before.
So I'm wondering if there are any papers/...
- 708
2
votes
0
answers
78
views
$\sigma$-fields as closure systems
Let $(\Omega,\mathcal A, P)$ be a probability space and let $\Sigma(\mathcal A) \subset 2^{\mathcal A}$ be the collection of all sub-$\sigma$-fields of $\mathcal A$. Then, $\Sigma(\mathcal A)$ is ...
- 1,731
2
votes
0
answers
56
views
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)) \...
- 5,407
2
votes
0
answers
117
views
Estimating the measure of a pre-image of a polynomial
This question was previously posted on MSE https://math.stackexchange.com/questions/3305781/estimating-the-measure-of-a-pre-image-of-a-polynomial
Let $\sigma := 2/(3\sqrt{3})$, be a real number. And ...
- 333
2
votes
0
answers
70
views
If $X^n$ is a sequence of càdlàg processes whose FDDs converge to a continous process $X$, does $X^n$ converge to $X$ in the Skorohod topology?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a complete locally compact separable metric space, $(X^n_t)_{t\ge0}$ be an $E$-valued càdlàg process on $(\Omega,\mathcal A,\...
- 167
2
votes
0
answers
61
views
Convergence to the probability generating function of a Poisson process
I'm working currently with a Poisson process trying to proove Renyi's Theorem, so far I want to show that
$\prod_{i=1}^{k_n}[z + (1-z)e^{-\mu(A_{n_i})}] \to e^{-(1-z)\mu(A)}$ as $\mu(A_{n_i}) \to 0$, ...
- 21
2
votes
0
answers
117
views
Probability bound involving random, convex sets
Let $X,Y$ and $Z$ be random vectors in $\mathbb{R}^d$ defined on $(\Omega,\mathcal{H},\mathsf{P})$ s.t. $Z$ is $\mathcal{F}$-measurable for some $\mathcal{F}\subset\mathcal{H}$. Define a family of ...
- 185
2
votes
0
answers
41
views
Distribution of a multivariate continuous process determined by that of linear combination of its coordinates?
To keep the question short: Let $C([0,1], \mathbb{R}^d)$, $d \geq 2$ be the space of all $\mathbb{R}^d$-valued continuous processes. $X$ and $Y$ are two $C([0,1],\mathbb{R}^d)$-valued random variates, ...
- 223
2
votes
0
answers
130
views
A question on probability measure on the unit ball of Banach spaces
Let $X$ be a Banach space and let $(x^{*}_{n})_{n}$ be a sequence in $X^{*}$. Suppose that $\sum_{n}|\langle x^{*}_{n},x\rangle |\leq \|x\|$ for all $x\in X$.
Question: Is there a probability measure ...
- 3,343
2
votes
0
answers
922
views
Isomorphism of probability spaces
Consider a surjective map $f:(X, \sigma(f)) \to (Y, \mathcal{Y})$, if a measure $\nu$ is given in $(Y, \mathcal{Y})$ the pullback $\nu(f(\cdot))$ is a measure on $(X, \sigma(f))$, similarly if $\mu$ ...
- 349
2
votes
0
answers
63
views
Sensitivity of a function against its random arguments
Let $g:R^{n+m} \to R$ be a deterministic function of some independent random variables $x_1,\ldots,x_n$ with distributions $f_{x_1}(x),\ldots,f_{x_n}(x)$ and some deterministic variables $z_1,\ldots,...
- 482
2
votes
0
answers
160
views
A construction of abstract Wiener spaces using Prokhorov's theorem
I am struggling with Leonard Gross's (original) construction of abstract Wiener spaces (AWS). His proof is somewhat convoluted, but from what I have been able to understand he constructs a certain ...
- 5,407
2
votes
0
answers
168
views
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