Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{\psi_2}\le K$, and let $H_d=(\mathbb R^n)^{\otimes_2 d}$. Let $L>0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\textsf{E}f(X^{\otimes d})\right\rvert>t \right\} \le C\exp\left[-c_K\mathcal I_{n,d}\left( \frac{t}{L n^{(d-1)/2}} \right)\right], \] where \[ \mathcal I_{n,d}(s)= \min\left\{ \frac{s^2}{d^2}, \frac{s^2}{d\log(e+nd/s^2)} \right\},\qquad s>0, \qquad \mathcal I_{n,d}(0)=0. \] The first rate is forced by changes in $\|X\|$. The second comes from changes of $X$ when its norm is nearly fixed. The proof constructs one coupling that controls both the coordinatewise conditional displacement and the mean squared Euclidean distance, and combines these bounds with a second-order estimate for $x\mapsto x^{\otimes d}$. The rate is minimax sharp, scale by scale, even when the subgaussian norms are bounded by an absolute constant. For bounded coordinates the logarithm in the second rate disappears.
Let $d\geq 4$ and let $R>0$. When $d=4$, assume that $R^2\in\mathbb{N}\setminus 4\mathbb{N}$; when $d\geq 5$, let $R^2\in\mathbb{N}$ be arbitrary. We prove the fixed-radius estimate $$\|A_R f\|_{\ell^{p'}(\mathbb{Z}^d)}\leq C_{d,p,\varepsilon}R^{-d(2/p-1)+\varepsilon}\|f\|_{\ell^p(\mathbb{Z}^d)}$$ for $(d+2)/d\leq p\leq 2$, where $p'$ is the H\"older conjugate exponent of $p$ and $A_R$ is the probability average over the lattice sphere of radius $R$. This extends the fixed-radius estimates of Kesler--Lacey and Hughes to the sharp lower endpoint $p=(d+2)/d$.
Let $d\geq5$. For a strictly increasing sequence $(\mu_k)$ of positive integers, set $\lambda_k=\mu_k!$ and consider the lacunary discrete spherical maximal operator $A_\star f:=\sup_k |A_{\lambda_k}f|$ associated with the discrete spherical averages \[ A_\lambda f(x):=\frac1{s_\lambda}\sum_{\substack{n\in\mathbb Z^d,\ |n|^2=\lambda}} f(x-n), \] where $s_\lambda:=\#\{n\in\mathbb Z^d:|n|^2=\lambda\}$. Kesler, Lacey and Mena proved that $A_\star$ is bounded on $\ell^p(\mathbb Z^d)$ for every $p>1$ if $\log\mu_k/\log k\longrightarrow\infty$, and asked about its endpoint behavior at $\ell\log\ell$. We resolve this endpoint question by characterizing all factorial sequences for which the $\ell\log\ell$ estimate holds. Define \[ C_{\log}=\sup_{N\geq2}\frac{\#\left\{k\geq 1:\mu_k\leq N\right\}}{1+\log N}. \] We prove that the $\ell\log\ell$ endpoint estimate holds if and only if $C_{\log}<\infty$. More precisely, if $C_{\log}<\infty$, then for every $\alpha>0$ and every finitely supported $f:\mathbb Z^d\to\mathbb C$, \begin{align*} \#\{x\in\mathbb Z^d:A_\star f(x)>\alpha\} \leq C_d(1+C_{\log})\sum_x \frac{|f(x)|}{\alpha} \left(1+\log^+\frac{|f(x)|}{\alpha}\right), \end{align*} where $C_d$ depends only on $d$. Conversely, if the above inequality holds with a finite constant $C_0$ in place of $C_d(1+C_{\log})$, then $C_{\log}\leq C_d(1+C_0)$. Thus their growth condition alone is insufficient at this endpoint. In particular, the estimate holds for $\lambda_k=(2^k)!$ and fails for $\lambda_k=(k+\lceil\exp(\sqrt{k})\rceil)!$.
Sanghyuk Lee, Ji Li, Chong-Wei Liang et al.· 0 citations
We fix an integer $d\ge1$ and a symmetric positive-definite matrix $Q\in\mathbb{R}^{d\times d}$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be finite, set \[ Z_\mu:=\int_{\mathbb{R}^d}e^{-V(x)}\,dx\in(0,\infty), \qquad d\mu(x):=Z_\mu^{-1}e^{-V(x)}\,dx, \] and assume that $\mu$ has finite second moment and that \[ x\longmapsto \frac12\langle Qx,x\rangle-V(x) \] is convex. Let $\nu$ be a compactly supported log-concave probability measure with support $K$, and let $\nabla\Phi$ be the Brenier map from $\mu$ to $\nu$. For $v\in\mathbb{R}^d$, define \[ w_K(v):= \sup_{y\in K}\langle y,v\rangle - \inf_{y\in K}\langle y,v\rangle. \] We prove that \[ \partial_{vv}\Phi \le 0.587 \sqrt{\langle Qv,v\rangle}\,w_K(v) \qquad(v\in\mathbb{R}^d) \] in the sense of distributions. We show that $\nabla\Phi$ has an everywhere-defined globally Lipschitz representative such that \[ \operatorname{Lip}(\nabla\Phi) \le 0.587 \sqrt{\|Q\|_{\mathrm{op}}}\,\operatorname{diam}(K). \] The directional Hessian estimate is affinely covariant, whereas the global Lipschitz estimate is dimension-free. The result also applies to singular or lower-dimensional targets. In particular, it removes the $\sqrt d$ loss in Kolesnikov's estimate for the Brenier map from Gaussian measure to normalised Lebesgue measure on a convex body. We also prove new bounds that depend only on the support for compactly supported semi-log-concave targets, which includes targets with bounded negative curvature.
A nonuniform version in which the failure probability depends on the individual parameters, and a lower bound showing that a universal constant prefix discrepancy is impossible when $d=o(\ln T)$, are proved.
Let $X=(X_1,\ldots,X_n)$ be independent nonnegative random variables, not necessarily identically distributed. Let $D=(D_0,D_1,\ldots,D_n)\sim\operatorname{Dir}(1,\ldots,1)$ be independent of $X$, and define $K(x)=\mathbb{P}\{\sum_{i=1}^n x_iD_i\le1\}$. We prove that, for every $n\ge1$, whenever $\mathbb{E} X_i\le1$ for every $i$, $\mathbb{P}\{K(X)\le\alpha\}\le\alpha$ for all $0\le\alpha\le1$. Thus $K(X)$ is a finite-sample, distribution-free $p$-value for testing the null hypothesis $\mathbb{E}X_i \le 1$ for all $i$. This proves a conjecture of Gaffke (2005).
Let $X\subset \mathbb C^n$ be a closed pure $d$-dimensional complex analytic set. We associate to $X$ its set of projective asymptotic directions $$ \Sigma_\infty(X) :=\{\ell \in \mathbb P^{n-1}(\mathbb C):\ell\cap C_\infty(X)\ne\{0\}\}, $$ where $ C_\infty(X)$ is the total tangent cone at infinity. We prove the metric Chow-type characterization $$X\text{ is algebraic} \quad \Longleftrightarrow \quad \mathcal{H}^{2d} \bigl( \Sigma_\infty(X)\bigr)=0. $$ As an application, if $C_\infty(X)$ is definable in an o-minimal expansion of $(\mathbb R,+,\cdot)$ and $\dim_{\mathbb R} C_\infty(X)\le 2d$, then $X$ is algebraic.
Nhan D. V. Nguyen· 1 citation· ⚡1
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