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$\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\la …

$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$The answer is yes. Indeed, it is easy to see (cf. e.g. Proposition 7 or the beginning of its proof) that the Wasserstein distance between …

This is to show rigorously that the uniform rectangular grid does not work -- cf. the answer by mike. As in the answer by Dieter Kadelka, suppose that the $cn$ vertical lines and the $cn$ horizontal l …

$\newcommand\Z{\mathbf Z}\newcommand\ep\varepsilon\newcommand\tP{\tilde P}\newcommand\de\delta\newcommand\R{\mathbb R}$Note that the random point $\Z/|\Z|$ is uniformly distributed on the sphere $S^{n …

Without loss of generality, $R=1$. Let $Z_1,\ldots,Z_n$ be iid standard normal random variables (r.v.'s). Then \begin{equation} \sqrt n\, X_1\overset{\text{D}}=\frac{\sqrt n\,Z_1}{\sqrt{Z_1^2+\cdot …

$\newcommand{\R}{\mathbb{R}} \newcommand{\x}{\mathbf{x}} \newcommand{\X}{\mathbf{X}}$ This is to present a formalization of the answer by Carlo Beenakker, without explicit use of the delta function. …

$\newcommand{\De}{\Delta}\newcommand{\R}{\mathbb R}\newcommand{\ga}{\gamma}\newcommand{\Ga}{\Gamma}$This is to provide a detalization on Will Sawin's comment. Specifically, let us show that the best u …

A quick way to generate a random point uniformly distributed in a bounded region $D$ is to generate a random point $P$ uniformly distributed in a rectangle $R$ containing $D$ and, if $P\notin R$, then …

Without loss of generality, the three points on the unit circle are $1,e^{is},e^{it}$, where $s$ and $t$ are uniformly and independently distributed on the interval $[0,2\pi]$, so that $a=|e^{is}-1|$, …

The intuition in the comment by Ori Gurel-Gurevich appears to be correct. Indeed, let us show that $\pi-\phi_n$ is on the order of $1/n^3$ in probability. Let $T$ denote the set of all triangles w …

Note that the random vector $G/\|G\|$ is uniformly distributed on the unit sphere in $\mathbb R^n$, where $G$ is a standard Gaussian random vector in $\mathbb R^n$. So, given an $n$-tuple $(x_1,\dots, …

$\newcommand\bar\overline$ Let us first find asymptotics of the radial distribution of a random point uniformly distributed on the standard/probability simplex $$S_n:=\{(x_1,\dots,x_n)\in[0,\infty)^n\ …

The answer is no, and the additional support/non-discreteness condition is of no help. Consider e.g. the following modification of the counterexample in Matt F.'s comment. Suppose that $P(U_m=0)=1/m=P …

As Nate Eldredge pointed out in his comment, your two-step procedure will not in general simulate (a random element of $\mathcal M$ with) the uniform density $f=1/|B_x|$ on $B_x:=\{y\in\mathcal M\colo …

This approach is of course well known. Clearly, it just says that $$P(X\in A)=P((N_1,\dots,N_d)\in C_A),$$ where $A$ is a Borel subset of the unit sphere $S^{d-1}$ and $C_A:=\mathbb R_+A$ is the corr …