17 problems

Minimal sparsity conjecture. The condition

Weak-dependence extension conjecture. An extension of the multivariate results to the other type of weak dependence discussed in the paper should also be possible.

An incomplete U-statistic is formed from a kernel whose dependence structure is represented by a deterministic design and whose asymptotic normality is established under the assump…

Let $U_n(f)$ be the classical $U$-statistic, and let $U_n^\circ(f)$ and the alternating $U$-statistics be the variants considered in the paper. In the degenerate cases, consider th…

Let $U_n$, $U^\circ_n$, $U^{-+}_n$, $U^{+-}_n$, and $U^{--}_n$ be the cyclic and alternating $U$-statistics considered in the paper, with their corresponding normalized variables.…

Let $u_2(X)$ denote the toy U-statistic considered in the paper, and let general U-statistics be formed from high-dimensional data. The bootstrap estimates discussed in the paper c…

Let a heavy-tailed multivariate average be approximated by a Gaussian distribution, and consider a carefully chosen sequence of Euclidean balls as the measurable sets on which the…

Let $U_{n,m,q}$ be the full two-sample U-statistic and let $U_{n,m,q}^M$ be its monotonically indexed counterpart, defined from two independent samples $X_1,\ldots,X_n$ and…

Let $U_{n,q}^{M,(qs)}$ denote the $(qs)$-th Hoeffding component of the monotonically indexed statistic $U_{n,q}^M$, and let $\boldsymbol\Sigma_s(q)$ be the corresponding covariance…

Consider the setting of Theorem Trate: a constrained or unconstrained U-statistic built from the random sequence and kernel in the paper, with Kolmogorov distance $d_K$ and the the…

Let $(X_i)_{i=1}^{\infty}$ be a stationary $m$-dependent sequence of random variables with values in a measurable space, let $\ell\geqslant1$, and let $f$ satisfy condition…

Let $B_n$ denote the degenerate term in the decomposition of the statistic under consideration, in the case where the statistic is univariate and ${\operatorname{Var}} s_x=0$. A mi…

Dependence-structure conjecture. This conditional-asymptotic approach should be possible if a suitable dependence structure allowing, for example, for big-block-small-block argumen…

Let a base learner predict by randomly selecting and averaging over points in a set $A$, and let $k$ denote the number of selected points. Write $\zeta_s$ and $\zeta_1$ for the cor…

Let $(X_1,X_2)$ be a random vector with an arbitrary distribution on $\mathbb{R}^2$, and let $h_{\tau^*}$ be the symmetric kernel defining Bergsma–Dassios–Yanagimoto's rank correla…

Let $W$ be a degenerate $U$-statistic of order $d$ in the setting of Theorem, and let $D_n$ denote the quantity appearing in that theorem and in Corollary. Conjecture on removing…

Optimal-rate conjecture. The optimal rate for $\rho(\bar{T}_n,\bar{Y})$ in the high-dimensional setting is