All Questions
24 questions with no upvoted or accepted answers
8
votes
0
answers
422
views
Non-affine smooth transformation of Gaussian is Gaussian
Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...
- 399
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 ...
- 1,124
6
votes
0
answers
388
views
Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
- 1,095
4
votes
0
answers
160
views
Can we show equivalence of two distributions based on their statistics?
Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \...
- 57
4
votes
0
answers
95
views
Approximating martingales given marginal distributions
Let $(\mu_0,\mu_1)$ be a vector of probability measures on $\mathbb R$ that are of finite first moment, i.e.
$$\int_{\mathbb{R}}|x|\mu_i(dx)~<~+\infty \mbox{ for } i=0,1$$
and increasing in ...
- 1,835
4
votes
0
answers
1k
views
Total variation and Hellinger distance inequality between truncated Gaussians
We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, $...
- 41
4
votes
0
answers
867
views
For what sub-$\sigma$-algebra are these two measures equivalent?
In two statistics papers (linked inline below) I have come across two definitions of certain probability measures. I conjecture that for particular choices of the construction that they are ...
- 2,791
3
votes
0
answers
179
views
Probability terminology
This is strictly a low-level terminology question. If I have a probability space $\Omega$ and a measurable space $S$, then a random variable $X:\Omega\rightarrow S$ gives rise via pushforward to a ...
- 23k
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
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
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
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
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
60
views
A canonical example of the non-existence of predictive probability distribution
Section 3 of Fortini et al. (2000) states that
Given $(X^\infty, \mathcal X^\infty,P)$, a predictive probability distribution of $x_n$ given $(x_1, \dots, x_{n-1})$ with respect to $P$ need not ...
- 131
2
votes
0
answers
160
views
Is it possible to improve the order of convergence of averages of random variables if they are not identically distributed?
Let $X_n$ be a sequence of independent random variables (but not necessarily identically distributed)
taking values in $[-1,1]$ that have the following property:
1) The average $A_n := \frac{(X_1+ \...
- 3,245
1
vote
0
answers
43
views
Does the constrained Wasserstein barycenter admit a blue noise property?
Let $(E,d)$ be a metric space and $\nu$ be a probability measure on $\mathcal B(E)$. In this paper, it is mentioned that sampling from $\mu$ can be described as choosing $n\in\mathbb N$, $x_1,\ldots,...
- 167
1
vote
0
answers
74
views
Measurability of $\mathbb{R}^n$-Random Field
Let $(X_x)_{x\in [0,1]^d}$ be a collection of integrable random variable defined on a (common) probability space $(\Omega,\mathcal{F},\mathbb{P})$. Under what condition is the map:
$$
[0,1]^d\ni x \...
- 5,405
1
vote
0
answers
105
views
Measure on a set and its value on $\emptyset$
After my first post here, I have one more doubt which is bothering me. It concerns Minlos's book Introduction to mathematical statistical physics again. To fix the notation, we have $\Lambda \subset \...
- 1,305
1
vote
0
answers
67
views
Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$
Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
- 19
1
vote
0
answers
96
views
Random projection increases the distance?
Consider two absolutely continuous random variables $X: \Omega \mapsto \mathbb{R}^d$ and $Y: \Omega \mapsto \mathbb{R}^d$ for probability spaces $(\Omega, \mathcal{F},p_X)$ and $(\Omega, \mathcal{F},...
- 482
1
vote
0
answers
146
views
approximation of probability distribution
I have a question: Let $\mu$ be a probability distribution defined on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ satisfying
$$\int_{\mathbb{R}}|x|d\mu<+\infty$$
Set
$$A_n=\Big\{\frac{i}{n}:~ i\in\...
- 1,835
0
votes
0
answers
93
views
Regularity with respect to the Lebesgue measure through dimensions
Let us consider two probability measures $\mu \in \mathcal{P}(\mathbb{R}^{p})$ and $\nu \in \mathcal{P}(\mathbb{R}^{q})$ with $p,q \in \mathbb{N}^{*}$. We note $\#$ the push forward operator i.e for $...
- 407
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
votes
1
answer
76
views
transformation of two measures on different space
Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$.
Let $\Sigma_U$ be the smallest $\sigma-$ field ...
- 1