0
votes
0answers
30 views

Uniform distribution on convex hull

Let $X=\{ x_1,\dots,x_n \} \subset \mathbb{R}^m$. Let $H(X)$ be the convex hull of $X$. Assume that $X$ is a convexly independent set, i.e. none of the $x_i$ are a convex combination of the others. ...
3
votes
1answer
34 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
2
votes
1answer
57 views

Solving $ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$

Let $\phi \in \mathcal ( [0,1]^2)$ symetric , can we find a solution to the following minimisation problem? $$ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$$ with $$ ...
1
vote
0answers
40 views

How to show this converges in probability

$A_n(s)$ is a sequence of convex random functions defined on an open set $S\in \mathbb{R}^p$ which converges in probability to some $A(s)$ for each $s$. I'm trying to show that $\sup_{s\in K} \big| ...
1
vote
0answers
41 views

Does Jensen's inequality become stricter with respect to the right boundary point?

Let $f(x)>0$ for all $t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ be a density. I will truncate the density to the finite interval $[a,b]$ and will eventually be taking ...
3
votes
0answers
94 views

What are the extreme points of the set of probability measures on $(\mathbb N, \mathcal{F})$?

Say, we have some $\sigma$-algebra on $\mathbb N$ and let $\mathbb P$ be the set of all probability measures on it. We know that $\mathbb P$ is convex, so I wonder how do the extreme points look like. ...
2
votes
1answer
21 views

Conxex combinations of max and min

Is the following true? $$α \left( \max_{p\in P}\int g\mathrm dp\right)+\left (1-\alpha \right ) \left( \max_{q\in Q}\int g\mathrm dq \right )=\max_{z\in\left (\alpha P+\left(1-\alpha \right )Q ...
1
vote
2answers
141 views

The distribution of barycentric coordinates

Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{R}^d$ and let $Z$ be a random variable uniformly distributed over convex hull of $\mathcal{X}$, denoted as $\text{conv}(\mathcal{X})$. Assuming ...
1
vote
2answers
68 views

Integral of convex set

Let $D$ be a convex set and $X_1,\dots,X_d$ be integrable random variables. If $X= (X_1,\dots,X_d)$ is in $D$ almost surely, why is it true that the vector $a= (\mathbb{E}X_1,\dots,\mathbb{E}X_d)$ ...
2
votes
1answer
186 views

Von Neumann's minimax theroem and Carathéodory's theorem

In J.F. Mertens(1986)(Paywall), there's a neat proof of a version of Von Neumann's minimax theroem. But I can't understand how Carathéodory's theorem is invoked. Suppose, in a two-person zero sum ...
2
votes
1answer
107 views

equivalent condition for moment generating function

Consider a random variable $x$ with pdf $f(x)$, and have $x \ge 0$. The moment generating function is defined as $M(t)=\int^{\infty}_{-\infty}e^{-tx}f(x)dx$ (noted that we change the sign of $t$ ...
5
votes
1answer
141 views

Taking a convex hull does not increase a supremum of a linear function

Let $X$ be a topological vector space, let $f:X\to\Bbb R$ be a continuous linear function and let $P(X)$ denote the set of all Borel probability measures on $X$. For any $M\subseteq X$ we define the ...
2
votes
1answer
217 views

An Orlicz norm is a norm

I had asked a question pertaining to Orlicz norms here. However, in the book I was reading, it said (and I paraphrase) "It's not difficult to show it is a norm on the space of integrable random ...
2
votes
1answer
122 views

Bound on Expectation of a convex function of a Random variable

My friend asked me the following question, which I at first thought was simple and straightforward: If $X$ is an integrable random variable and $g$ is a convex function(all real valued), then is it ...
10
votes
2answers
177 views

What is the expected convex depth of a set of $m$ randomly chosen points in the unit square?

Definition. Let $X$ be a set of points in $\mathbb{R}^2$. Then the vertex sequence of $X$ is defined by $X_{0}=X$, and $X_{i+1}=\left\{x\in \operatorname{Conv}(X_{i}): x \notin ...
0
votes
0answers
178 views

Distribution of convex combination of i.i.d Gamma random variables

I am wondering what one can say regarding the convex combination of i.i.d Gamma random variables? Specifically, consider $x_{i}$ be $Gamma(\theta,1)$, then would we have the following and if yes, ...
2
votes
1answer
95 views

Continuous and non-decreasing but how?

I am reading a paper and the author shows the continuity and monotonicity of a function. It seems so simple to see but I am sorry that I couldnt see the reason. I will be very happy if you can point ...
2
votes
3answers
116 views

“Interpolation” of polynomials

I'm dealing with a probability problem and I have to understand the following operation on polynomials: let $F$ and $G$ be any two polynomials of variable $p\in [0,1]$ (to be thought of as a Bernoulli ...
2
votes
1answer
106 views

Mathematical expectation is inside convex hull of support

Let $\xi$ be a random variable supported in some set $A \in \mathbb{R}^n$: $\xi \in A$ a.e. How to show that $\mathsf E \xi \in \mathop{\mathrm{conv}}{A}$? Let $s(x)$ be a support function of set ...
3
votes
1answer
179 views

Anderson's Inequality for Gaussian measures

Let $C\subset \mathbb R^n$ be convex and symmetric about the origin. I am trying to prove that $\gamma(C) \geq \gamma(C+x)$ for any $x\in \mathbb R^n$, where $\gamma$ is the standard Gaussian measure. ...
1
vote
0answers
88 views

Geometry of log-concave density functions and its distribution

Suppose $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is log concave (density function). Consider now the antiderivative (distribution function) $F(t)=\int_{{ x\le t}}f(x)dx$, which is also log concave. We ...
1
vote
0answers
77 views

Application of Anderson's theorem to probability

From Wikipedia, Anderson's theorem is stated as: Let $K$ be a convex body in n-dimensional Euclidean space $\mathbb{R}^n$ that is symmetric with respect to reflection in the origin, i.e. $K = ...
1
vote
0answers
35 views

Uniqueness of function representation as a mean value

Let $f(x)$ be a smooth function from $\mathbb{R}^n_{+}$ to $\mathbb{R}_{+}$ (it may be analytic). Are there some sufficient conditions on such function $f$ such that $$ f(x_1 y_1, \ldots, x_n y_n) ...