17 problems

Let $W$ be the normalized sum considered in the paper, let $\kappa_j(W)$ denote its $j$-th cumulant, and let $f\in\mathcal{C}^{k,\omega}(\mathbb{R})$, where $k\in\mathbb{N}_+$ and…

Let $x_1>x_2>\cdots>x_N$ be the symmetric ground state particle locations satisfying … Let the empirical distribution of $\{x_n:n=1,\ldots,N\}$ be compared with the standard Gaussi…

Let $X$ and $Y$ be independent non-centered normal random variables, and let $W=XY$. A second-order Stein operator for $W$ is a differential operator acting on smooth functions wit…

The paper studies several asymptotic estimates whose current error terms are weaker than order $n^{-1/2}$; the relevant results are Theorems … , and … , … should hold with error te…

Let $X\sim N(0,1)$), and let $Y=h(X)$, where $h\in\mathbb{K}[X]$ is a polynomial with $\deg(h)\geq 3$ and ${\rm I\kern-0.16em E}[h(X)]=0$. For $p_0\in\mathbb{K}_0[Y]$, let…

Let $\alpha\in(1,2)$, and consider approximation of a $d$-dimensional stable law by the relevant normalized sums. The approximation rate is measured in $L^{1}$ or total variation d…

Stein-factor gap conjecture. For all $k>\lambda$,

Let $X_1,X_2,\ldots$ be independent identically distributed random variables with symmetric Pareto tails of index $\alpha\in(0,1]$, and let $S_n=n^{-1/\alpha}\sum_{i=1}^n X_i$. The…

Stein universality lemma. For every $h\in\mathcal{H}$, this equation admits a bounded solution $f_h$ that is $d$ times differentiable, with

Let $X_1,\ldots,X_m$ be i.i.d. random variables and $Y_1,\ldots,Y_n$ be i.i.d. random variables, with $\mathbb{E}X_1=\mathbb{E}Y_1=0$, $\mathbb{E}X_1^2=\mathbb{E}Y_1^2$, and finite…

Let $r>0$ and $\lambda>0$ be the shape and rate parameters of the Gamma distribution, and let $f_h$ be the solution of the Gamma Stein equation for a test function $h$ with bounded…

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…

Let $T_\lambda$ be the power divergence statistic for goodness-of-fit across $n$ trials and $r\geq 2$ classifications, with index parameter $\lambda\in\mathbb{R}$. For a smooth tes…

Let $n_v$ denote the number of vertices and let $\lambda$ be the parameter governing the expected subgraph counts. In the regime considered, the two Skellam parameters satisfy … Th…

Let $f_x$ be the solution considered in the Skellam approximation, and define … For parameters $\lambda_1,\lambda_2>0$ with $\lambda_1\ne\lambda_2$, bounded-difference conjecture.…

Let $g(x;\theta)$ be a parametric family of densities with parameter $\theta$. Assume that $g(x;\theta)$ satisfies the regularity conditions required in the source. Fix a value…

Let $Z$ be a random variable with density $p$ satisfying assumptions $\textbf{(A1)}$ and $\textbf{(A2)}$, and let $\tau$ be related to $p$ by the defining relation in the paper. Fo…