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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)...
De vinci's user avatar
  • 399
4 votes
1 answer
1k views

General version of Skorokhod representation of random variables

Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the ...
arjun's user avatar
  • 941
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 ...
CodeGolf's user avatar
  • 1,835
3 votes
1 answer
304 views

Question abouth Skorokhod representation of random variables

It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e. $$\rho(\mu,\nu)<\varepsilon,$$ then there exist two random ...
CodeGolf's user avatar
  • 1,835
2 votes
1 answer
838 views

Does $\int_{\mathbb R^d} (1+|x|^{1 + \alpha}) \ell (x) \, d x < \infty$ imply $\int_{\mathbb R^d} (1+|x|) |\ell (x)|^{1-\alpha} \, d x < \infty$?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ We fix $\alpha \in (0, 1)$. Let $\ell : \bR^d \to \bR_+$ be a continuous function such that $$ \|...
Akira's user avatar
  • 835
2 votes
2 answers
600 views

Uniqueness of the limit sequence of discrete probability measures

Let $N_n$ be a sequence of natural numbers increasing to infinity, and suppose we have a sequence of finite sets of distinct points $X_n = \{x_1^{n},x_2^{n},\ldots,x^{n}_{N_n}\} \subset[0,1] \subset \...
Ben Ciotti's user avatar
2 votes
1 answer
231 views

Is Boltzmann entropy well-defined for arbitrary probability density function?

$\newcommand{\bR}{\mathbb{R}}\newcommand{\diff}{\mathop{}\!\mathrm{d}}$ We define a continuous function $\varphi : \bR_+ \to \bR$ by $$ \varphi (s) := \begin{cases} 0 &\text{if} \quad s =0 , \\ s \...
Akira's user avatar
  • 835
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 ...
Ritwik's user avatar
  • 3,245
2 votes
0 answers
174 views

Product of marginals absolutely continuous with respect to a Borel probability measure

Let $\mu$ be a Borel probability measure on $\Bbb{R}^{m+n}=\Bbb{R}^m\times\Bbb{R}^n$. Consider its marginal measures $\mu_1(A):=\mu(A\times\Bbb{R}^n)\, (A\in\mathcal{B}(\Bbb{R}^m))$ and $\mu_2(B):=\mu(...
KhashF's user avatar
  • 3,599
2 votes
0 answers
192 views

Convergence of Gibbs distribution to Dirac measure [closed]

Consider the probability density function on $R^d$ for a continuous function $F: R^d \to R$: $$ q_{\varepsilon}(x) = \frac{1}{Z} \exp\left(-\frac{1}{\varepsilon} F(x)\right). $$ Denote $x^* = \arg \...
test-account's user avatar
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,...
Jeff's user avatar
  • 482
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+ \...
Ritwik's user avatar
  • 3,245
1 vote
1 answer
166 views

Question abouth Skorokhod representation of random variables (II)

This is a continuation of Question abouth Skorokhod representation of random variables Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that $$\int_{\mathbb R}|x|^pd\mu(x),~ \...
CodeGolf's user avatar
  • 1,835
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
0 votes
0 answers
146 views

Derivatives in unusual support domains

Originally posted on Math.StackExchange, here, but I was advised to post it on MathOverflow as it is a research question. Now two final, great answers have been posted, see on Math.StackExchange. I ...
Vincent Granville's user avatar