10 problems

Let $\mathbf{W}^{(n)}=\mathrm{W}_2^2(\hat\mu^{1,(n)},\hat\nu^{1,(n)})$ be the squared Wasserstein distance between two empirical samples of size $n$ from the standard normal distri…

Let $P_{MXY}$ be the joint distribution of random vectors $M$, $X$, and $Y$. Define the … -PID is also Gaussian, so the $_G$-PID is identical to the … -PID rather than only the sta…

Best-fit Stiefel tropical linear space conjecture. As $\sigma\to0$, the quantity

Monotonicity conjecture. The expectation above monotonically decreases with $k$. Consequently, as $\sigma\to0$, the hyperplane that best fits $X$ converges to $H_0$, meaning that t…

Let $I$ be the AUK-based dependence index for a random vector, and let…

Let $a>0$ and consider approximating a Gaussian distribution by probability distributions supported on $[-a,a]$. Truncation conjecture. For large $a$, the almost optimal choice is…

Let $X$ be a smooth projective variety, and consider the sequences formed by the ranks of the syzygies associated with $X$. Gaussian limit conjecture for syzygy ranks. These sequen…

Cantelli's conjecture. The random variable $X+varphi(X)\cdot Y$ has a Gaussian distribution if and only if $varphi$ is constant.

Let $(X_1,X_2,\ldots,X_n)$ be a set of gaussian variables arising from several independent $n$-tuples of $d$-level particles impinging on different detectors. For each detector…

Gaussian dilatation conjecture. The sequence $(\alpha_n^{\theta^{\star},m})_{n\geq 1}$ is asymptotically $L^s(P)$-optimal for every $s>0$. Numerical experiments for the one-dimensi…