We establish a power-sum convergence principle for randomly weighted means. Let $P(t)=(P_j(t))$ be random finitely supported subprobability weight sequences, independent of iid centered integrable marks. If the expected total mass converges and the expected power sum of some order $r\in(0,1)$ is uniformly bounded, then convergence, for one fixed mark law, of $\sum_jP_j(t)X_j$ to a nondegenerate law forces $\mathbb{E}\sum_jP_j(t)^p$ to converge to a positive limit for some $p\in(1,2]$. The proof combines normal-family compactness for Mellin transforms of characteristic-function remainders, inversion at frequencies $\pm u$, and Landau's theorem at the abscissa of convergence; a uniform Abelian estimate handles the endpoint $p=2$. For weights obtained by normalizing a Poisson-sized iid sample of nonnegative variables, a gap below one for the logarithmic slope of the Laplace exponent yields the required power-sum bound of order below one. A Poissonized ratio Tauberian theorem then identifies the common tail of the unnormalized variables as regularly varying, with index $-\beta$ for a unique $\beta\in[0,1)$. As an application, this proves the remaining necessity direction in Breiman's 1965 conjecture for centered integrable marks. Combined with Breiman's sufficiency theorem, it completes the conjecture.
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 consider sums of freely independent self-adjoint random variables that are not necessarily identically distributed. Let $\mu_j$ denote the distribution of the $j$th summand. We assume that they have mean zero and finite absolute moments of order $2+\delta$, where $0<\delta\le 1$. Let $\Delta$ denote the Kolmogorov distance, let $\mu^{(n)}$ be the distribution of the normalized partial sum, let $\omega$ be the standard semicircle law, and let $B_n^2$ be the variance of the partial sum. The purpose of this paper is to prove the Berry--Esseen estimate in the free central limit theorem. Namely, there exists an absolute constant $C>0$ such that, for every $0<\delta\le 1$, \[ \Delta(\mu^{(n)},\omega) \le \frac{C}{B_n^{2+\delta}}\sum_{j=1}^n \int_{\R}|x|^{2+\delta}\,\mu_j(dx), \] Our result not only improves several known estimates for general non-identically distributed random variables, but also establishes exactly the same Berry--Esseen estimate as in classical probability theory. The proof combines truncation, a quantitative estimate for the $R$-transform, a stability analysis of a perturbed semicircle equation, and a Bai-type smoothing inequality.
We give upper and lower bounds for the number of solutions of the equation $e_n(x,y) = g$ in the group $W_k=(C_p\wr C_{p^k})^2$, where $e_n(x,y)$ is the $n$-th Engel word and $g\in W_k$. We obtain several corollaries from this. First, we prove a stronger version of the Amit-Ashurst conjecture for Engel words in $W_k$. We also prove that Engel words are not probabilistic identities in profinite groups with arbitrarily large wreath product quotients $W_k$. To conclude, we construct closed subsets of $(C_p\wr\Z_p)^2$ with positive Haar measure, empty-interior, and which are the preimage of an Engel word map.
Let $A_n$ be a subset of $\{1,2,\ldots,n\}$ obtained by retaining each integer independently with fixed probability $\theta\in(0,1)$, and let $L_n$ be the least common multiple of the integers in $A_n$. We prove a functional large deviation principle, a functional moderate deviation principle, and a Strassen-type functional law of the iterated logarithm for the process $(\log L_{\lfloor{nt}\rfloor})_{0\le t\le1}$. The large deviation rate function is given by an entropy contraction for geometric marks, while the moderate deviation rate function and LIL cluster set are described by the reproducing kernel Hilbert space associated with the Gaussian limit process.
Shaochen Wang, Guangyu Yang, Wang Zhou· 0 citations
We establish a central limit theorem in R\'enyi divergence for independent and identically distributed lattice random variables $X_1, \cdots, X_n$ with zero mean, unit variance, and maximal span $h>0$. Let $S_n=(X_1+\cdots+X_n)/\sqrt n$. Let $Z_n$ denote the standard Gaussian distribution quantized on the support lattice of $S_n$. For every $\alpha>1$, with $\beta=\alpha/(\alpha-1)$, we prove that the R\'enyi divergence $D_\alpha(S_n\|Z_n)\to 0$ if and only if the divergence is finite at some convolution level and the strict sub-Gaussian condition $$ \mathbb E e^{tX}<e^{\beta t^2/2},\quad t\in\mathbb R,~ t\ne0 $$ holds. Under these conditions, we further derive an Edgeworth-type asymptotic expansion of the divergence to arbitrary order. These results provide a lattice counterpart of the R\'enyi entropic central limit theorem for continuous random variables due to Bobkov, Chisyakov and G\"{o}tze (\emph{Ann. Probab.} \textbf{47} (2019), 270--323).
Let $X_1,X_2,\ldots$ be independent $\mathrm{Bernoulli}(\theta)$ random variables, and let $\bar X_n = n^{-1}(X_1 + \cdots + X_n)$. We prove that, for every real $p \geq 1$, the sequence $\{\mathsf{E}(\bar X_n^p)\}_{n \geq 1}$ is log-convex. This proves the Bernoulli case of a conjecture of Lamkin and Tkocz [Canad. Math. Bull., 65(2):271-278, 2022]. The proof conditions on the total number of successes among $2n$ trials and reduces the desired inequality to a convex-order comparison between two normalized quadratic functions of hypergeometric random variables. All the log-convexity inequalities are strict for $p>1$ and $0<\theta<1$.