Tagged Questions

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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. ...
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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 ...
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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 ...
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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 $a= (\mathbb{E}X_1,\dots,\mathbb{E}X_d)$ ...
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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 ...
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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 $t$ ...
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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 ...
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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 ...
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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 ...
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 $X_{i+1}=\left\{x\in \operatorname{Conv}(X_{i}): x \notin ... 0answers 176 views Distribution of convex combination of i.i.d Gamma random variables I am wondering what one can say regarding the convex combination of i.i.d Gamma random variables? Specifically, consider$x_{i}$be$Gamma(\theta,1)$, then would we have the following and if yes, ... 1answer 95 views Continuous and non-decreasing but how? I am reading a paper and the author shows the continuity and monotonicity of a function. It seems so simple to see but I am sorry that I couldnt see the reason. I will be very happy if you can point ... 3answers 116 views “Interpolation” of polynomials I'm dealing with a probability problem and I have to understand the following operation on polynomials: let$F$and$G$be any two polynomials of variable$p\in [0,1]$(to be thought of as a Bernoulli ... 1answer 106 views Mathematical expectation is inside convex hull of support Let$\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)$be a support function of set ... 1answer 179 views Anderson's Inequality for Gaussian measures Let$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$is the standard Gaussian measure. ... 0answers 87 views Geometry of log-concave density functions and its distribution Suppose$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$, which is also log concave. We ... 0answers 77 views Application of Anderson's theorem to probability From Wikipedia, Anderson's theorem is stated as: Let$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 = ...
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) ...