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Is finding the CDF from the Laplace transform well-posed?

In my study of Dynamic Light Scattering, I came across the following inverse problem. Let $F(s):[0,T]\rightarrow[0,T]$ be the Laplace transform of a probability distribution $f(t)$ on the real line ...
Riemann's user avatar
  • 654
-1 votes
1 answer
168 views

Space of distributions on $[0,1]^2$: weakly compact or not?

Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$. Question 1: Does $\mathcal{X}$ ...
tom jerry's user avatar
  • 359
5 votes
1 answer
375 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(...
user5034's user avatar
11 votes
1 answer
677 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 ...
Tutukeainie's user avatar
4 votes
2 answers
354 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 $$ Tf=f*p.$$ ...
Ribhu's user avatar
  • 407
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 \...
MathLearner's user avatar
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 $\...
shawn532's user avatar
1 vote
1 answer
150 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 $\...
Akira's user avatar
  • 825
3 votes
1 answer
219 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, ...
Analyst's user avatar
  • 657
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 ...
jack412's user avatar
  • 63
0 votes
1 answer
328 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 $[0,M]$ (...
Grandes Jorasses's user avatar
3 votes
1 answer
100 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 ...
user141240's user avatar
0 votes
1 answer
105 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 ...
edgar314's user avatar
1 vote
0 answers
64 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 ...
BabaUtah's user avatar
  • 149
0 votes
1 answer
296 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 $...
edgar314's user avatar
15 votes
0 answers
477 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 ...
Dor's user avatar
  • 723
9 votes
1 answer
359 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 $(f(x_i+x_j))_{i,j=1}^n$...
Hans's user avatar
  • 3,031
1 vote
0 answers
96 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 \...
user490373's user avatar
10 votes
1 answer
330 views

(Sharp) Bounds on $E(XYZ)$ given all the bivariate marginals

Suppose $X,Y,Z$ are all real-valued random variables. Suppose I know the joint marginal distributions of $(X,Y)$, $(Y,Z)$ and $(X,Z)$. I want to find bounds on $E(XYZ)$. In the case of bounding $E(XY)$...
Atom Vayalinkal's user avatar
6 votes
2 answers
333 views

Is there a way to reconstruct the convolution $(f * g)(x)$ of $f$ with a Gaussian $g$ from sampled values, $(f*g)(a), a \in A$?

Suppose that $f: \mathbb{R} \to \mathbb{C}$ is a function which has support in $[-1,1]$. Let $g = g_\sigma$ be a centered Gaussian with variance $\sigma^2$. Is there a way to reconstruct the ...
J. Swail's user avatar
  • 437
4 votes
0 answers
656 views

Eigenvalues of Matérn covariance function

Recall that Matérn covariance function $C_\nu(d)$ is defined as $$ C_\nu(d)=\sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{\rho}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{\rho}\right), ...
Zuofeng Shang's user avatar
12 votes
1 answer
1k views

Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces

Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
Matheus Manzatto's user avatar
4 votes
3 answers
251 views

A functional integral inequality

Suppose $f:I=(0,1)\to \mathbb R$ is a continuous function that satisfies $$ \int_I f(t) e^{at}\,dt \geq 0\quad \text{for all $a \in \mathbb R$}.$$ Does it follow that $f\geq 0$ on $I$?
Ali's user avatar
  • 4,153
9 votes
1 answer
652 views

Scaling in Mehta's integral

The following expression is known as Mehta's integral and deeply connected to random matrix theory: $$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
Pritam Bemis's user avatar
3 votes
1 answer
984 views

About the metrizability of the space of Probability measures $\mathcal{P}(S)$

It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the ...
vaoy's user avatar
  • 309
7 votes
1 answer
1k views

Properties of convolutions

Consider the function $$f_{n}(x)=e^{-x^2}x^n.$$ and the function $$h_p(x):=e^{-\vert x \vert^p}.$$ My goal is to analyze $$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*...
Landauer's user avatar
  • 173
0 votes
0 answers
84 views

Determining the tails of a convolution from its behavior on a compact set

Let $p$ be a smooth (say, $C^\infty$, but this is not crucial) density on the interval $I=[0,1]$ and $g_\sigma$ be the density of $N(0,\sigma^2)$. Define $f=p\ast g_\sigma$. To what extent does the ...
user13322's user avatar
0 votes
1 answer
307 views

Regularity properties of conditional distributions

Let $(X,Y)\in\mathbb{R}^n\times\mathbb{R}^m$ be a pair of random variables with joint density $p(x,y)$. I am interested in the regularity properties of the conditional densities $p(y|x)$ and $p(x|y)$ (...
user19200's user avatar
0 votes
1 answer
1k views

Bounding $L^p$ norms in terms of lower-order $L^q$ norms

Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
JohnA's user avatar
  • 710
3 votes
1 answer
404 views

The sign of the tail of Fourier transform of a positive function/ characteristic function

I am interested in a specific density (positive function) and would like to prove that the tail of its characteristic function (Fourier transform) is positive ($>0$). Here is the density $f(x)=c_\...
Tanya Vladi's user avatar
14 votes
0 answers
718 views

Lower bounds on analytic functions connected to Fox H

The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
Tanya Vladi's user avatar
1 vote
1 answer
115 views

$K(x,y)\in L^{\infty}(R^n\times R^n, m\times m)$, $K(x,y)=K(y,x)$, so $K(x,y)=\sum_{k=1}^{\infty}\lambda_k \phi_k(x)\phi_k(y)$, are $\phi_k$ bounded?

Consider a symmetric function $$ K(x,y):R^n \times R^n \to R $$ satisfying $K(x,y)=K(y,x)$ and $$ \int_{R^n\times R^n} K^2(x,y)dm(x) dm(y) <\infty. $$ Let $m$ be a probability measure on $R^n$. ...
mathmetricgeometry's user avatar
3 votes
1 answer
299 views

Lipschitz functions that saturate the Lipschitz inequality on the average (part 1)

Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality \begin{align*} |f(x) - f(y)| \le \|x-y\|_2, \;\forall x,y \in \mathbb R^n. \end{align*} For $n \ge 2$, can we ...
passerby51's user avatar
  • 1,731
1 vote
1 answer
369 views

Does the almost sure convergence of absolutely continuous r.v.'s imply the weak convergence of the pdf's in $(L^\infty)^*$?

The following question was asked in a comment at Almost sure convergence vs convergence of probability density functions : Suppose that $(X_n)$ is a sequence of random variables (r.v.'s) converging ...
Iosif Pinelis's user avatar
4 votes
1 answer
1k views

Simple proof of Prékopa's Theorem: log-concavity is preserved by marginalization

The following result is well-known: Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have $$H \left( (1 - \lambda)(x_1,y_1) + \...
Sascha's user avatar
  • 536
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 $L^1(\mathbb{R}^d)$ ...
neverevernever's user avatar
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\...
Jeff S's user avatar
  • 75
0 votes
1 answer
212 views

Expressing the measure of a set in terms of the characteristic function of the measure

Let $\mu$ be a discrete, finitely supported probability measure in $\mathbb{R}^d$ and denote by $\phi$ be the characteristic function of $\mu$, i.e. $\phi(t)=\mathbb{E}e^{i<t,X>}$, where $X$ is ...
TOM's user avatar
  • 2,288
2 votes
2 answers
155 views

Existence of classical solution for a parabolic equation without Hölder continuity in time for its coefficients

Consider equation $$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$ with initial condition $u(0, x) = g(x).$ Suppose that $c(t, x)$ and $...
kenneth's user avatar
  • 1,399
1 vote
0 answers
65 views

Normalizing constants preserve metric entropy

Suppose $\mathcal{F}=\left\{f\in L^2([a,b]): 0<\underline{c}\leq f\leq\overline{c} \right\}$. Consider the following transformation $$\tilde{\mathcal{F}} := \left\{\frac{f}{\int f d\mu}: f\in \...
lucaszz's user avatar
  • 11
5 votes
2 answers
415 views

Existence of Solution, System of Equations

Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$ I think the following system of equations ...
TikoM's user avatar
  • 53
3 votes
1 answer
315 views

Where to find the proof of this property?

I am doing some exercises in the analytic and there is a problem as following: ``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that: $\sum\limits_{n=1}^{+\infty} f_n = 1$. $\...
mathJuan's user avatar
  • 153
3 votes
2 answers
265 views

Can one realize this as an ergodic process?

Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph. We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$ In other words: For ...
user avatar
5 votes
1 answer
170 views

Ratio of integrals with increasing dimension over Euclidean balls

Let $f_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int_{\mathbb{R}^{n}}f_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\...
neverevernever's user avatar
1 vote
0 answers
56 views

Moduli of continuity and Wasserstein differentiability of functions between measures

Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
JeffHolder's user avatar
2 votes
0 answers
189 views

Point wise convergence of Laplace transform and convergence of functions

Assume that functions $f_n(t), f(t)\in C_b(R_+)$. For every $\lambda >0$, we have $$ \bigg|\int_0^\infty e^{-\lambda t}f_n(t)d t-\int_0^\infty e^{-\lambda t}f(t)d t\bigg|\leq C_\lambda n^{-1}, $$ ...
Wenguang Zhao's user avatar
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 ...
BigbearZzz's user avatar
  • 1,245
2 votes
1 answer
268 views

Monotonicity of the Hellinger integral/distance

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and $\...
Iosif Pinelis's user avatar
2 votes
1 answer
263 views

Schwartz space on $\bigcup_{n=1}^CR^n$

I have an application where I need to work with the following idea. Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...
Thiru's user avatar
  • 21
7 votes
1 answer
624 views

Expectation involving maximum of Gaussian variables

Let $X\sim N(0, I_d)$ be a $d$-dimensional Gaussian random vector. Let $W_1, \ldots, W_k \in \mathbb{R}^d$ be $k$ fixed vectors in general positions. It is clear that $w_i^\top X, \ldots, w_k^\top X$ ...
Steve's user avatar
  • 1,127