All Questions
69
questions
5
votes
1
answer
285
views
Looking for a counterexample: Conditioning increases regularity?
Let $p(x,y,z)$ be a joint density (over $\mathbb{R}^3$) under no smoothness or regularity assumptions, besides its existence. I am looking for a (counter)example where $p(y|x)$ is less regular than $p(...
- 51
2
votes
0
answers
88
views
Dependence and $L^2$ projections of functions
tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function?
Let $w$ be a density on $\...
- 21
4
votes
2
answers
330
views
Injectivity of a convolution operator
Let $p,\mu,\nu$ be probability density functions on
$\mathbb{R}$ such that
$$
\int_{\mathbb{R}}p(y-x) \nu(y) \, dy=\mu(x).
$$ Now, consider the operator $T:L^2(\mu)\to L^2(\nu)$ such that $$ Tf=f*p.$$ ...
- 361
11
votes
1
answer
607
views
Entropy arguments used by Jean Bourgain
My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...
- 196
1
vote
1
answer
137
views
Is the Boltzmann entropy continuous in the supremum norm?
We define $U : [0, +\infty) \to [0, +\infty)$ by $U(0) := 0$ and $U (s) := s \log s$ for $s >0$. Then $U$ is strictly convex. Let $D$ be the set of all bounded non-negative continuous functions $\...
- 1,179
3
votes
1
answer
190
views
Is there a real/functional analytic proof of Cramér–Lévy theorem?
In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment
The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...
- 647
3
votes
1
answer
97
views
Vague Topologies induced by $C_c$ and $C_0$ are the same on a closed ball of finite Radon measures?
Let $X$ be a locally compact Hausdorff space. Denote $C_c(X)$ and $C_0(X)$ the space of continuous functions with compact support and vanishing at infinity respectively. By Riesz representation ...
- 305
1
vote
0
answers
59
views
Sequential Hölder-norm for functions in $H_{\alpha}([0,1]^{d})$?
I have come across a nice result attributed to Ciesielski (Ciesielski, Z. (1960). On the isomorphisms of the spaces $H_{\alpha}$ and m. Bull. Acad. Pol. Sci. Ser. Sci. Math. Phys. 8, 217–222.), even ...
- 11
0
votes
1
answer
306
views
Deduce that a function is zero on interval $[0,M]$
I have been thinking about this for the last few days but I was not able to produce a definitive answer.
Take an integrable function $g$ that maps in $\mathbb{R}$ and with domain contained in $[0,M]$ (...
- 131
0
votes
1
answer
97
views
Transforming two smooth densities to the same density
I am looking for an example of the following:
Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the ...
- 5
1
vote
0
answers
94
views
Building random homeomorphisms of the circle
Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as
\...
- 101
5
votes
2
answers
2k
views
Relationship between KL, chi-squared, and Hellinger
There are many well-known relationships between the KL divergence, chi-squared ($\chi^2$) divergence, and the Hellinger metric. In the paper "Assouad, Fano, and Le Cam" by Bin Yu, the author ...
- 63
0
votes
1
answer
288
views
When can a convolution be written as a change of variables?
Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
Under what conditions does $X=h(Y)$, where $...
- 5
15
votes
0
answers
472
views
Quantitative Skorokhod embedding
The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
- 723
9
votes
1
answer
344
views
Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
- 2,927