13 problems

Consider two Gaussian random walks and an empirical kurtosis statistic formed from their paths. The numerator and denominator determining the limiting law of $\theta_n$ converge in…

Let $u_N(t,x)$ be the truncated Wiener-chaos expansion … and let $T_p(\theta)$ be the transition time defined in the paper. For $p\geq 2$, write $\|u_N(t,x)\|_p$ for the $L^p$-norm…

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 $F_\infty$ be the target random variable in the second Wiener chaos of the form referenced in the source, and let $F=I_q(f)$ be a chaotic random variable in the $q$th Wiener ch…

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

Let $\nu>0$, and let $F=I_2(f)$ belong to the second Wiener chaos with … Let $G(\nu)\sim\operatorname{CenteredGamma}(\nu)$. Two-sided total variation conjecture. There exist consta…

Total variation rate conjecture. In this setting, the estimate

Let $\{F_n\}_{n \ge 1}$ be a sequence of random variables in a fixed Wiener chaos or Wigner chaos of order $p\ge 2$, normalized by … for all $n\ge 1$. In the classical case, let…

Let $(X_1,X_2,X_3,X_4)$ be a jointly Gaussian vector, and consider the two-dimensional vector whose components are sums of two Gaussian squares, $(X_1^2+X_2^2,X_3^2+X_4^2)$. Infini…

Let $\phi$ be the three-dimensional BMS field sampled according to the non-Gaussian measure $\nu_{\phi}$. A field is subordinated to a Gaussian field with finite Wiener chaos repre…

Fix $q\geq 2$. Let $\{F_n\}_{n\geq 1}$ be a sequence with $F_n=I_q(f_n)$, where $f_n\in \mathcal{H}^{\odot q}$, and suppose that … Let $Y$ have the symmetric Gamma distribution…

Let $X$ be a multiple Wiener--Itô integral of order $p\geq 2$, and let $\kappa_6(X)$ denote its sixth cumulant. Sixth-cumulant positivity conjecture. For every such $X$, one has ……

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…