Let $X$ be a centered isotropic log-concave random vector in $\mathbb{R}^n$. For $\theta\in S^{n-1}$, let $\mu_\theta$ be the law of $\langle X,\theta\rangle$, and let $\Theta$ be uniformly distributed on $S^{n-1}$, independently of $X$. We prove the sharp estimate \[ \textsf{E} W_1(\mu_\Theta,\gamma_1) \le \frac{C}{n}. \] Here the Wasserstein distance is computed after the direction is fixed and is then averaged over the sphere. No symmetry assumption is imposed. A product measure with centered exponential coordinates gives a matching lower bound of order $n^{-1}$. The proof separates the averaged-direction law from the fluctuation among fixed directions. For the first part, a Taylor expansion in the random radius retains a mean-zero cancellation and yields an $O(n^{-1})$ error. For the second, a weighted $L^2$ distance between distribution functions is converted into an exact spherical kernel depending only on $|x|^2$, $|y|^2$, and $\langle x,y\rangle$. Expanding this kernel in $\langle x,y\rangle$, we control its linear, quadratic, and cubic terms using the quadratic variance inequality $\operatorname{Var}(X^\top M X) \le 8\operatorname{Tr}(M^2)$, while fixed-order moment estimates control the remainder.
Let $W=n^{-1/2}\sum_{i=1}^n X_i$, where the $X_i$ are independent centered random vectors in ${\mathbb R}^p$ with $|X_{ij}|\le B$ almost surely. Suppose that $\text{Cov}(W)$ has unit diagonal and smallest eigenvalue at least $b^2>0$. We prove that the distance between $W$ and a Gaussian vector with the same covariance, uniformly over axis-aligned rectangles, is at most $C\min\{1,b^{-2}Bn^{-1/2}\log^{3/2}(ep)\}$. For fixed $b$, the dependence on summand size and dimension matches known lower bounds in growing-dimensional regimes. The proof combines a concentration estimate near rectangle boundaries with a carefully chosen Gaussian comparison.
Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In particular, this resolves a question of E.M. Stein from the mid 1990s. The principal ingredient in our proof is that, when $t\lesssim d$ with $t$ sufficiently large, the associated multiplier $\mathfrak{m}_{\sqrt{\lfloor t^2\rfloor}}(\xi)$ admits an asymptotic expansion of arbitrary prescribed order, uniform in $\xi$, whose resulting maximal operators can be controlled by the discrete normalized Gaussian maximal function studied by Mirek--Szarek--Wr\'obel \cite{MSW25}.
Let $\{Y_n; n\ge 1\}$ be a sequence of independent and identically distributed random variables with mean zero in Peng's framework of the sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$, and $S_n=\sum_{i=1}^nY_i$. In this paper, we establish a limit law of \begin{align*}\lim_{n\to \infty}\max_{k\le n}\frac{S_k}{\sqrt{2k \log\log n}}. \end{align*} Different from the result obtained by Chen (2015) in which the limit is a constant, it is shown that under the upper capacity the limit may be prescribed as a given function of $Y_1,Y_2,\ldots$, taking values in the standard deviation interval. As a result, it is also shown that the set of limit points in the compact law of the iterated logarithm can be a symmetric random interval. This paper (Chinese version) has been submitted to Special Issue of Science in China-Mathematics in Celebration of Professor Peng Shige's 80th Birthday. In Theorem 2.2 of the original paper, an additional condition (2.6) is needed.
Li-Xin Zhang, Yongze Song· Scientia Sinica Mathematica· 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 $R$ be an $m\times m$ correlation matrix satisfying $R-\mathbf{1}\mathbf{1}^{\mathsf T}/m\succeq0$, let $X\sim\mathcal{N}(0,R)$, and let $Z_1,\ldots,Z_m$ be independent standard Gaussian random variables. We prove $\max_i X_i\leq_{\mathrm{st}}\max_i Z_i$, with equality in distribution if and only if $R=I_m$. We use this comparison to resolve the Weak Simplex Conjecture: among $d+1$ equiprobable equal-energy signals in $\mathbb{R}^d$ transmitted over an additive white Gaussian noise channel, the regular simplex is the unique maximizer of the average probability of correct maximum-likelihood decoding at every signal-to-noise ratio. The same comparison proves the Simplex Mean Width Conjecture and gives the exact finite-energy performance of deterministic no-feedback AWGN codes with equiprobable messages, no restriction on the number of channel uses, and a maximal per-codeword energy constraint. The proof uses a Gaussian product inequality for log-concave functions whose first moments with respect to standard Gaussian measure vanish. A variational argument chooses one exponential tilt and one truncation endpoint in each coordinate so that this product inequality applies and a Gaussian change of measure returns all coordinates to the prescribed common threshold. A strict form of the product inequality also shows that, unless $R=I_m$, $\mathbb{P}\{X\leq c\mathbf{1}\}>\Phi(c)^m$ for every finite $c$, and hence gives the distributional equality statement. A Lean formalization is available at https://github.com/abhmul/weak-simplex-conjecture-lean.
We confirm the Kannan--Lov\'asz--Simonovits conjecture for quadratic forms: if $X \sim \mu$ is an isotropic log-concave random vector in $\mathbb{R}^n$, then for any symmetric matrix $M$ one has $$ \operatorname{Var}_{X \sim \mu}(\langle MX,X\rangle) \leq 2\,\mathbb{E}_{X \sim \mu}|\nabla\langle MX,X\rangle|^2. $$ As an application, we apply the above to $M=\mathbb{E}_{X \sim \mu}(\langle X,\theta\rangle X\otimes X)$ for $\theta\in S^{n-1}$ and show that the Kannan--Lov\'asz--Simonovits constant $\psi_n$ satisfies $$ \psi_n\leq C\log^{1/4}n $$ for some absolute constant $C>0$.
B. Letwin· 10 citations· ⚡8
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