All Questions
34 questions
4
votes
2
answers
389
views
Gaussian mixtures are dense in total variation?
Let $M_{TV}(\mathbb{R}^d)$ denote the set of probability measures on $\mathbb{R}^d$ with finite total variation norm which are absolutely continuous with respect to the Lebesgue measure.
By a Gaussian ...
- 5,405
1
vote
1
answer
100
views
Does convergence of Radon transforms of a sequence of probability distributions implies convergence of the distributions themselves?
Let $P_1,P_2,\ldots $ be a sequence of absolutely continuous probability measures on $\mathbb R^n$, and let
$f_j:\mathbb R^n\to\mathbb R$ be their PDFs. Assume that $\operatorname{E}P_j = 0$ and $\...
- 13
0
votes
0
answers
128
views
When is the image of $T \colon \ell^2 \to \ell^2$ a Gaussian random variable?
In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable.
Now suppose that $x$ is a (say, centered) ...
- 420
0
votes
2
answers
101
views
Minimal set of functions to characterize a distribution
In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by:...
- 109
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 ...
- 5
1
vote
1
answer
241
views
Integration by parts for indicator of a sphere to indicator of a ball
Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
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 $...
- 5
2
votes
0
answers
69
views
A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?
Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI)
$$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$
with LSI constant $\...
- 21
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 ...
- 19
1
vote
1
answer
141
views
Does the compactness of parameter of distribution function imply the compactness of the distribution (or probability measure) in Wasserstein space?
For a family of probability measures sharing the same form of distribution function $F(x; p)$ with different parameters (i.e., $p$'s), if the parameter falls in a compact subset of real line, can we ...
- 13
0
votes
1
answer
479
views
Probabilistic interpretation of derivative of a Dirac delta function
Consider $g : \mathbb{R}^d \mapsto \mathbb{R}$ defines some surface $\Sigma$ in $\mathbb{R}^d$. Then I can define a random variable $X_1$ with support only on $\Sigma$ by using a pdf of the form $$p_1(...
- 333
2
votes
0
answers
168
views
A slight generalization of Skorokhod's representation theorem
Let $f:\mathbb{R}^p\rightarrow\mathbb{R}^q$ $(p,q\geq 1)$ be a continuous function and $(X_n)_{n\geq 1}$ a sequence of random values on $\mathbb{R}^p$ such that $f(X_n)$ converges in law to a random ...
- 449
0
votes
0
answers
144
views
Optimization over the set of all bounded probability measures
Given $X$ finite, fix a continuous function $\theta \in \Delta^+ (X) \to [0,1]$, fix a probability measure $\mu^*$, and a $\varepsilon > 0$. Consider:
$$
\max_{\mu \in \Delta^+ (X)} \theta (\mu), \...
- 67
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
1
answer
1k
views
Functional derivative of differential entropy
I have trouble finding the derivative of the differential entropy w.r.t the probability density function, i.e. what is $\frac{\delta F[p]}{\delta p(x)}$, where $F[p] = \int_X p(x)\ln(p(x))dx$, and $p(...
- 41
3
votes
1
answer
983
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 ...
- 309
0
votes
2
answers
210
views
Limited sum for whole sum approximation
Let $d_n, n\in\{1,2,\cdots,N\}$ be $N$ realizations drawn independent and identically from uniform distribution on $(0,L)$ where $L=\gamma\sqrt{N}$ with constant $\gamma$. Suppose that we need to ...
- 287
1
vote
2
answers
234
views
Find $\inf_{P_{X_1,X_2}}P_{X_1,X_2}(\|X_1-X_2\| > 2\alpha)$ , where $\alpha > 0$ and inf is over couplings
Let $\mathcal X$ be a seperable Banach space with norm $\|\cdot\|$, and let $X_1$ and $X_2$ be random vectors on $\mathcal X$ with finite means.
Question. Given $\alpha > 0$, what is value of, ...
- 6,853
2
votes
1
answer
436
views
Best approximation of a compactly supported density by a single Gaussian
Note: This is a follow-up question inspired by a previous (more difficult) question I asked on MathOverflow.
Let $f:\mathbb{R}\to\mathbb{R}$ be a (sufficiently regular, e.g. smooth) probability ...
- 710
1
vote
0
answers
56
views
About a class of expectations
Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...
- 2,246
2
votes
1
answer
266
views
A question about finite free convolution
For any square matrix $Y$ let $\chi_x(Y) = det(xI -Y)$ denote its characteristic polynomial.
Say $A$ and $B$ are two $n-$dimensional symmetric matrices with constant row sums $a$ and $b$. Lets ...
- 2,246
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
1
vote
1
answer
183
views
Diffuse measure space as a product of $[0;1]$ and another diffuse measure space
The title speaks of itself. How far is an arbitrary finite diffuse measure space from being almost isomorphic to a product of $[0;1]$ with another diffuse measure space? What would be reasonable ...
- 1,959
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
2
votes
1
answer
401
views
Reference on Probability theory on functional spaces (in special Hilbert spaces)
Currently, I am working on some sort of stochastic optimization problems defined over function spaces.
I am familiar with standard probability theory (R. Durrett, ''Probability: Theory and Examples")...
7
votes
1
answer
719
views
Tightness and Functional Analysis
Let $(\Omega , \mathbb{P})$ be a probability space and $X$ be a real-valued random variable. Then we immediately have the push-forward measure $\mu$ on $\mathbb{R}$ and one can think of $\mu$ as an ...
- 2,283
2
votes
0
answers
619
views
Laplace transform of a integral function of CIR/CEV process
The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...
- 323
1
vote
1
answer
125
views
A differential inequality and a special value
Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq 2{G'(x)}^2.$$...
- 11.3k
0
votes
1
answer
221
views
Behavior of the integral of products of probability densities
Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition
$$
T(x_1,\ldots,x_n) := \...
- 65
6
votes
3
answers
2k
views
Estimating the variance of a discrete normal distribution
Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...
- 167
2
votes
1
answer
354
views
star-product of copulas
I have recently come accross the star product of copulas, that is if $A$ and $B$ are 2-copulas and $\{C_t\}_{t\in[0,1]}$ is a family of copulas, then $C(x,y,z) = \int_0^y C_t(\frac{\partial}{\partial ...
- 23
0
votes
1
answer
229
views
Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
- 125
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 3,245
-1
votes
1
answer
696
views
Can singular measures be viewed as vanishing distributions? (Answer No!)
Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
- 1,649