For any finite family of Poisson multiple integrals and any finite $p\geq2$, we construct a common increasing filtration generated by finitely many exact Poisson counts such that the associated conditional expectations converge in $L^p$, remain in their original chaoses, and have bounded step kernels with finite-measure support. This finite-count martingale core allows regular fixed-chaos identities and estimates to be extended under the sole assumption of a finite fourth moment. In particular, if $F$ lives in a Poisson chaos with unit variance and finite fourth moment, we prove that the Kolmogorov distance between $F$ and a standard normal is bounded by $15.6(\mathbb{E}[F^4]-3)^{1/2}$. This removes Assumptions $\mathbf A$ and $\mathbf A^{\textbf{loc}}$ from the Kolmogorov bound of D\"obler and Peccati (Ann. Probab., 2018). We also obtain quantitative $L^4$ estimates for all iterated Malliavin derivatives and, for $F$ in a Poisson chaos, the fourth moment assumption of $F$ forces the $L^4$-integrability of its kernel.
Let $(F_n)$ be a sequence of random vectors with identity covariance matrix whose components belong to the same fixed Wiener chaos, and assume that $F_n$ converges in law to a standard Gaussian vector. We prove that, for every integer $m\geq0$ and every $p\in[1,\infty]$, the optimal rate of convergence of the densities in the Sobolev space $W^{m,p}$ is given by the maximum of the absolute third-order cumulants and the diagonal fourth-order cumulants. The same quantity also gives the optimal rates in total variation, Kolmogorov and $1$-Wasserstein distances. Our proof first derives, by Gaussian interpolation and Gaussian convolution, a cumulant expansion in the space of tempered distributions without imposing Malliavin nondegeneracy at the endpoint. Finite-order Malliavin density estimates then upgrade this identity to Sobolev spaces. Matching lower bounds follow from a finite-dimensional argument in which parity separates the third-order and fourth-order Gaussian corrections, while norm equivalence rules out cancellations among mixed cumulants. A superconvergence theorem supplies the required finite negative moments of the Malliavin determinant along a sufficiently far tail of the approximating sequence, so that no Malliavin nondegeneracy assumption is required.
We develop Gaussian approximation bounds in higher-order Wasserstein distance $W_p$, $p\geq2$, for sums of multivariate martingale differences generated by a uniformly ergodic Markov chain. Under an $L^{(2+\eta)p}$-moment condition with $\eta>0$, we establish the explicit bound $$ O\left( p^3 \|A\|_4^2 + pd^{1/4}\|A\|_2^{1/2}\|A\|_4^2 \right) $$ where $A\in\mathbb{R}^n$ collects the $L^{(2+\eta)p}$-sizes of the $n$ individual martingale increments. In the balanced-increment regime where the individual increments have comparable sizes of order $n^{-1/2}$, it yields the first optimal $O(n^{-1/2})$ Gaussian approximation rate for fixed $p$ and $d$. Consequently, we also obtain the first optimal $O(n^{-1/2})$ $W_p$ Gaussian approximation rate for multivariate additive functionals of uniformly ergodic Markov chains. Our analysis develops two techniques for addressing the interplay between higher-order Wasserstein distance and temporal dependence. First, building on the Ornstein--Uhlenbeck relative-score approach of Fang and Koike (2023), we formulate the bound in terms of antisymmetric Stein couplings while retaining the conditional tensor structure. Second, we develop a refresh-then-maximal coupling that combines an independent first-step resampling, which preserves the desired Stein identity, with a subsequent maximal coupling that provides effective control of the coupling increment. These tools may be useful more broadly for Gaussian approximation under temporal dependence.
We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every dimension $n\geq3$. We also give a six-term cubic example in four variables, which was found first and already proves failure for every $n\geq4$. Both examples follow from the same coefficient identity. The search was prompted by Levent Alp\"oge's public announcement of an explicit three-dimensional counterexample to the Jacobian Conjecture. Although the main theorem of Derksen, van den Essen, and Zhao is stated globally in dimension, its proof has fixed-dimensional content: a noninvertible cubic-homogeneous Keller map in $r$ variables forces the failure of ${\mathrm GMC}(2r)$. Tracking a standard Bass--Connell--Wright reduction of the announced map gives a conservative cubic-homogeneous counterexample in $79$ variables, and hence a route-based failure of ${\mathrm GMC}(158)$. That route is nonconstructive at the final Gaussian step and does not furnish explicit polynomials $P,Q$. The much smaller explicit failures in dimensions $4$ and $3$ below were not derived from the announced Jacobian map.
We study extremal absolute central moments of Gaussian quadratic forms under a fixed Frobenius norm. For a nonzero real symmetric matrix $M$ and $G\sim N(0,I_n)$, we construct an explicit centered difference of Gamma variables with the same mean, variance, and third centered moment as $G^{\mathsf T}MG-\operatorname{tr}M$. After normalization, replacing the quadratic form by this Gamma difference does not decrease $\mathbb{E} f$ for every $C^2$ test function $f$ such that $f''$ is convex and $f$, $f'$, and $f''$ have polynomial growth. In particular, it gives an explicit upper bound for every absolute moment of order $p\ge3$. We then prove that, for every $p\ge4$, this bound is maximized by the centered square $g^2-1$ of a single standard Gaussian variable $g$. The resulting sharp inequality is \[ \bigl\|G^{\mathsf T}MG-\operatorname{tr}M\bigr\|_p \le \|g^2-1\|_p\,\|M\|_{\mathrm F}, \] with equality if and only if $M$ has rank one.
We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_n\in \operatorname{Mat}_n(\mathbb{F}_p)$ has independent and $\alpha_n$-balanced entries. We prove that if $$ \liminf_{n\to\infty}\frac{n\alpha_n}{\log n}>1, $$ then, for every fixed collection of distinct monic irreducible polynomials over $\mathbb{F}_p$, the corresponding primary partitions of $A_n$ converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform model studied by Fulman in his thesis. The sharp sparsity threshold for the full rational canonical form coincides with the threshold previously obtained by Lee for finite-field cokernels and by Jung-Lee-Yu for random matrix models over $\mathbb{Z}_p$. Our proof is based on the surjection moment method over the function field $\mathbb{F}_p[t]$, applied to the finite module $\operatorname{Cok}_{\mathbb{F}_p[t]}(tI_n-A_n)$, whose primary decomposition records the rational canonical form. We also construct degree-$d$ critical sparse obstructions, suggesting a polynomial-dependent threshold $1/d$ for statistics associated with irreducible polynomials of minimal degree $d$.
We prove an optimal Hardy inequality for every aperiodic, symmetric random walk in $\mathbb{Z}^2$ with finite variance. In particular, we verify null-criticality, and thus, optimality of the underlying Hardy weight. Under suitable moment conditions, we use fine asymptotics of the potential kernel due to Fukai and Uchiyama in order to derive the asymptotics of the weight. For the standard Laplacian, we recover the expected first order term in the asymptotics but also show that next order term is negative. Thus, our result shows that the constant in the Hardy inequality proven by Kapitanski and Laptev cannot be larger than $1/4$, which is the optimal constant in the continuum. The proof of null-criticality rests on a new criterion for general graphs beyond the locally finite case. We also recover the situation of $\mathbb{Z}^d$ with $d \geq 3$ which can also be also treated by our new method.
Philipp Hake, Matthias Keller, Felix Pogorzelski· 0 citations
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