论文标题
当地的Glivenko-Cantelli
Local Glivenko-Cantelli
论文作者
论文摘要
如果$μ$是$ d $二维的布尔立方体$ \ {0,1 \}^d $的发行版，我们的目标是估算基于$ n $ iid draws in $μ$的[0,1]^d $中的平均$ p \。具体来说，我们考虑经验平均值$ \ hat p_n $，并研究预期的最大偏差$Δ_n= \ mathbb {e} \ max_ {j \ in [d]} | \ hat p_n（j）-p（j）-p（j）| $。在经典的通用glivenko-cantelli设置中，人们在$δ_n$上寻求无分配（即独立于$μ$）的边界。该制度非常理解：对于所有$μ$，我们都有$δ_n\ lyssim \ sqrt {\ log（d）/n} $的通用常数，并且界限很紧。 我们目前的工作旨在建立对$δ_n$的估计，包括$δ_n$的估计，包括$ d = \ infty $的估计。由于这样的界限一定必须取决于$μ$，因此我们将此体制称为{\ em local} glivenko-cantelli（也称为$μ$ -GC），并且意识到这种类型的以前的范围很少 - ``abraction''''''产品的特殊情况已经是$μ$的，这是不平凡的。对于$δ_n\ to0 $，我们提供了$μ$的必要条件，并计算此衰减的尖锐费率。一路上，我们发现了一种新颖的粉刺型次伽马型最大不平等，对伯诺利（Bernoullis）转移，具有独立的兴趣。
If $μ$ is a distribution over the $d$-dimensional Boolean cube $\{0,1\}^d$, our goal is to estimate its mean $p\in[0,1]^d$ based on $n$ iid draws from $μ$. Specifically, we consider the empirical mean estimator $\hat p_n$ and study the expected maximal deviation $Δ_n=\mathbb{E}\max_{j\in[d]}|\hat p_n(j)-p(j)|$. In the classical Universal Glivenko-Cantelli setting, one seeks distribution-free (i.e., independent of $μ$) bounds on $Δ_n$. This regime is well-understood: for all $μ$, we have $Δ_n\lesssim\sqrt{\log(d)/n}$ up to universal constants, and the bound is tight. Our present work seeks to establish dimension-free (i.e., without an explicit dependence on $d$) estimates on $Δ_n$, including those that hold for $d=\infty$. As such bounds must necessarily depend on $μ$, we refer to this regime as {\em local} Glivenko-Cantelli (also known as $μ$-GC), and are aware of very few previous bounds of this type -- which are either ``abstract'' or quite sub-optimal. Already the special case of product measures $μ$ is rather non-trivial. We give necessary and sufficient conditions on $μ$ for $Δ_n\to0$, and calculate sharp rates for this decay. Along the way, we discover a novel sub-gamma-type maximal inequality for shifted Bernoullis, of independent interest.
