8 problems

Let $X$ be an $n$-dimensional random vector, let $Z_t$ be an independent Gaussian random vector with covariance matrix $tI$, and set $X_t=X+Z_t$ for $t>0$. Write $H(X_t)$ for the d…

Let $\alpha$ be real, and let $a,b$ be coprime integers with $b\to\infty$ and $a/b\to\alpha$. Define $c_\alpha$ to be the variational quantity given in the source by the infimum of…

Let $u\ge 0$, $N_1,N_2\ge 0$, and let $\Sigma_1,A_2\succeq 0$. For random variables $X_i,Z_i$ in $\mathbb{R}^d$, where $d\ge 1$, suppose that … The variables $Z_1,Z_2$ are Gaussian…

Let $X$ be an $n$-dimensional random vector and define $X_t=X+Z_t$, where $Z_t$ is an independent Gaussian random vector with covariance matrix $tI$. Let $X_G$ be an $n$-dimensiona…

Let $X$ be an $n$-dimensional random vector, let $Z_t$ be an independent Gaussian random vector with covariance matrix $tI$, and set $X_t=X+Z_t$. Let $X_G$ be an $n$-dimensional Ga…

The Paris-14E Parc Montsouris dataset consists of daily average temperatures organized into patterns, and the Grassberger–Procaccia approach estimates intrinsic dimension from the…

Let $n$ be the sample size, let $s\in(0,2]$ be the smoothness parameter, let $d$ be the dimension, and let $\mathcal{H}_d^s(L;[0,1]^d)$ denote the Hölder class used in the minimax…

Entropy-region conjecture for three vector-valued Gaussians. The closure of $G_3^{\mathrm{Gauss}}$ is characterized by the following inequalities. If $\tilde y\leq 0$, then