All Questions
Tagged with pr.probability measure-theory
823 questions
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
0
votes
1
answer
133
views
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
5
votes
0
answers
537
views
Conditional probabilities in Banach spaces
This is the infinite-dimensional sequel to my question, Conditional probabilities are measurable functions - when are they continuous?.
Let $\Omega = \Omega_1 \times \Omega_2$ be a probability space ...
CommunityBot
- 1
2
votes
1
answer
272
views
Can a superadditive set function be upper-bounded by a measure?
Suppose $Q$ is a probability measure on a Euclidean space $\mathbf{E}$. Suppose $\mathcal{K}$ is the collection of all compact sets on $\mathbf{E}$.
Consider the function $f: \mathcal{K} \rightarrow ...
- 1,074
2
votes
1
answer
152
views
Computationally random bitstreams and normalcy
Let $\mathbb{N}$ denote the set of non-negative integers. We can identify every bitstream, i.e. a function $s:\mathbb{N}\to \{0,1\}$, with some $A\in{\cal P}(\mathbb{N})$: take $A = s^{-1}(\{1\})$.
...
- 54.2k
3
votes
2
answers
639
views
Lifting a probability measure to the power set
Let $X\neq\emptyset$ be a set and let $\mu:{\cal P}(X)\to [0,1]$ be a probability measure. Is there a probability measure $$\bar{\mu}:{\cal P}({\cal P}(X))\to [0,1]$$ with the following property?
...
- 12.3k
7
votes
1
answer
1k
views
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
11
votes
0
answers
381
views
Concerning Luzin-(N)-property
Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set.
By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
- 4,201
1
vote
1
answer
457
views
Taylor's theorem for a composition with $\min:\mathbb R^2\to\mathbb R$ and differentiability Lebesgue almost everywhere
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1$$
$g:=\ln f$ (and assume $g'$ is Lipschitz continuous)
$n\in\mathbb N$, $$s(x,y):=\sum_{i=1}^n\left(g(y_i)-g(x_i)\right)$$ and $$h(...
CommunityBot
- 1
5
votes
1
answer
1k
views
Sum of random variables are equal in distribution
Suppose that $X,Y$ are scalar random variables supported on some standard Lebesgue probability space $(\Omega, \mathrm{P})$, such that $X \overset{\mathrm{d}}{=} Y$ in the sense that their pushforward ...
- 14.2k
6
votes
1
answer
575
views
Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel
I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant.
Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
- 14.2k
19
votes
3
answers
1k
views
Which distributions can you sample if you can sample a Gaussian?
Explicitly: You have a computer that is able to pick a real number at random according to the normal distribution: $\mathcal{N}(0,1) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. Which distributions can this ...
- 6,045
7
votes
1
answer
261
views
Comparison of several topologies for probability measures
Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
- 7,514
1
vote
1
answer
211
views
Can we order random variables in a measurable way in a general setup?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E,\mathcal E)$ be a measurable space
$n\in\mathbb N$
$X_1,\ldots,X_n$ be $(E,\mathcal E)$-valued random variables on $(\Omega,\...
- 128k
4
votes
2
answers
667
views
Convergence (topology) for $\sigma$-finite measures
I'm having much trouble finding literature that addresses the questions which I write below. I was wondering if someone could help me out to understand better, either by providing references or by ...
- 29.8k
2
votes
1
answer
451
views
Show that the absolute value of this function is twice differentiable except on a set of Lebesgue measure $0$
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1\tag1$$
$g:=\ln f$ and assume that $g'=\frac{f'}f$ is Lipschitz continuous (note that this implies that $f'(x)\xrightarrow{|x|\to\...
- 16.2k
8
votes
1
answer
425
views
Measurability for disintegration of a kernel
Let $(x, A) \mapsto P(x, A)$ be a probability kernel whose "target" (wikipedia terminology) is a product space $Y \times Z$, and say both $Y$ and $Z$ are compact metric spaces. For every $x$ there is ...
CommunityBot
- 1
2
votes
1
answer
437
views
If $g$ is differentiable, how can we show that $z\mapsto1\wedge e^{g(z)}$ is differentiable except on a countable set
If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, ...
- 109k
3
votes
2
answers
926
views
Weak convergence of conditional probabilities
Suppose $\mu_n\implies\mu$, i.e. $\mu_n$ converges weakly to $\mu$ where $\mu_n$, $\mu$ are probability measures on some metric space $(X,d)$. Given a Borel set $B$, define $\mu^B$ to be the ...
- 87
1
vote
1
answer
690
views
Integrable version of the Borel-Cantelli theorem?
I have taken an introductory course on measure theory where I learned about the Borel-Cantelli theorem but I wonder whether there is a lebesgue integrable version. Given an uncountable collection of ...
- 3,871
2
votes
1
answer
235
views
Kolmogoroff condition for truncated random variables
Question summary. Does the Kolmogoroff condition $\sum_{n=1}^\infty\frac{\mathbb V Y_n}{n^2} < \infty$ hold for truncated random variables $Y_n := X_n \cdot 1_{\{X_n \le n\}}$ (see below for a more ...
- 128k
2
votes
2
answers
241
views
Substitute Concrete Value in Conditional Expectation
Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space.
Let $$ X, Y : \Omega \rightarrow \mathbb{R} $$ be random variables.
Furthermore, let
$$ f: \mathbb{R}^2 \rightarrow \mathbb{R} $$
be a $...
- 128k
1
vote
0
answers
163
views
Does the law of a Feller process depend continuously on the initial condition?
Let $E$ be a locally compact and separable metric space, and suppose $X$ is a Feller process with transition function $P_t$. To be precise, let $C_0$ denote the space of continuous functions vanishing ...
- 11
3
votes
0
answers
232
views
Research-type questions in probability illustrating measure-theoretical techniques for students
In short, in the perspective of preparing a new course, I am looking for examples of "concrete" (hopefully research-type) questions concerning various models in probability theory which give the ...
CommunityBot
- 1
8
votes
1
answer
355
views
Lower Bound of KL-Divergence Between Two Gibbs Measures
Suppose we have two Gibbs measures with densities
$$
p_f(x) \propto \exp(f(x)),\quad q_g(x)\propto \exp(g(x)).
$$
Consider the KL-divergence between $p_f$ and $q_g$, as a functional of $f$ and $g$, ...
- 128k
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
6
votes
1
answer
2k
views
Topological conditions of Kolmogorov Extension Theorem
KET is often used to construct stochastic processes in continuous time when the state space is $\Bbb R^d$. As far as I am familiar with its proof, it uses standard monotonic class-like arguments ...
1
vote
0
answers
192
views
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
2
votes
1
answer
2k
views
Convergence of probability density function
There are various kinds of (convergence of random variables) but I have never read about convergence of density functions.
Let $X_1, X_2, \dots, X$ be random variables $\Omega \to \mathbb{R}$ and $...
- 128k
4
votes
1
answer
432
views
Proving Conditional Independence
Each of the scalar random variables, $ Y $, $ X $, $ U $, and $ V $, is continuous and possibly has $ \mathbb{R} $ as its support. The random variable, $Z$, could be vector valued, but continuous.
I ...
- 128k
3
votes
2
answers
163
views
Questions about some properties of random probabilities and random expectations
Let $(\Omega, \mathcal{A}, \mathbb P)$ be a probability space with $\mathcal{A}$ countably generated, and let $P: \mathcal{A} \times \Omega \to [0,1]$ be a random probability measure. By that I mean $...
- 17k
0
votes
0
answers
424
views
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
vote
0
answers
81
views
If $\text P\left[X_2\in B_2\mid X_1\right]=\kappa(X_1,B_2)$ a.s. for all $B_2$, can we select a common null-set over all $B_2$?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E_i,\mathcal E_i)$ be a measurable space
$X_1:\Omega\to E_1$
$X_2:\Omega\to E_2$ be $(\mathcal A,\mathcal E_2)$-measurable
$\kappa$ ...
- 167
3
votes
2
answers
227
views
Example of measure for some algebra over N
$\mathcal F$ is set of events. Can you give an example of some algebra $\mathcal A$ over $\mathbb N$ and a non-zero finitely additive measure $\mu$ on this algebra, which has a countably additive ...
- 4,991
5
votes
2
answers
470
views
Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$
Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...
CommunityBot
- 1
8
votes
1
answer
171
views
On the existence of a particular type of finite measure on $\mathbb N$
Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: ...
- 12.8k
1
vote
0
answers
58
views
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
2
votes
1
answer
170
views
Measurability of C([0,1]) for the completion of the Wiener measure
Consider the completion $(\mathbb{R}^{[0,1]}, \mathcal{B}, \mu)$ of the Wiener measure on $\mathbb{R}^{[0,1]}$ (with the cylinder set $\sigma$-algebra).
Is the following true :
$C([0,1])\in \...
- 29.8k
4
votes
1
answer
412
views
Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator?
I have a basic question about Gaussian measures on a Hilbert space:
Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the ...
- 128k
4
votes
2
answers
415
views
Effect of perturbing the atoms of a measure on the Wasserstein distance
Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
- 576
5
votes
1
answer
386
views
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
2
votes
1
answer
326
views
What is the Wiener measure of the curves with Hölder index $\frac 1 2$?
One may show that the Wiener measure (for curves in $\mathbb R^n$) is concentrated on the Hölder-continuous curves of Hölder index $< \frac 1 2$. What happens to the curves of Hölder index ...
- 5,407
6
votes
1
answer
196
views
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
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
1
vote
0
answers
120
views
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
119
views
time delay ergodic theorem
given dynamic system $(X, \mathcal{B}, F, \mu), \mu \circ F^{-1}=\mu, F $ is mixing, $ A \in \mathcal{B}, s.t. \mu(A) >0 $.
consider dynamic system $(X\times X, \mathcal{B}\otimes \mathcal{B}, ...
- 23.2k
7
votes
1
answer
424
views
Transportation-cost inequality for pushforward measure
Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies ...
- 128k
4
votes
1
answer
2k
views
wasserstein distance between distributions with bounded ratio
Let $p$ and $q$ be probability distributions on a metric space $X=(X, d)$ with densities $dp$ and $dq$, such that there exists $0 < \alpha < \beta < \infty$ satisfying
$$
\alpha d p \le dq \...
- 6,853
3
votes
0
answers
198
views
Cardinality of extreme points of finitely additive probabilistic extensions
Let $\Omega = \{0,1\}^\mathbb{N}$, let $\mathcal{A}$ be the algebra generated by the open subsets of $\Omega$, where we use the product of discrete topologies, and let $\mathcal{F} = \sigma(\mathcal{A}...
- 869
5
votes
0
answers
696
views
Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)
Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail.
...
- 151