Le Cam spacings theorem in dimension two

关于间距m元组和函数的中心极限定理

The central limit theorem for sum-functions of m-tuples of spacings

该研究针对单位圆周随机点的均匀间距m元组函数和，在m随n增长的条件下推导了期望、方差的渐近公式并证明其渐近正态性，拓展了间距相关极限理论的适用范围。

Gaussian limits for generalized spacings

Nearest neighbor cells in $R^d,d\in\mathbb{N}$, are used to define coefficients of divergence ($ϕ$-divergences) between continuous multivariate samples. For large sample sizes, such distances are shown to be asymptotically normal with a variance depending on the underlying point density. In $d=1$, this extends classical central limit theory for sum functions of spacings. The general results yield central limit theorems for logarithmic $k$-spacings, information gain, log-likelihood ratios and the number of pairs of sample points within a fixed distance of each other.

A complement to Le Cam's theorem

This paper examines asymptotic equivalence in the sense of Le Cam between density estimation experiments and the accompanying Poisson experiments. The significance of asymptotic equivalence is that all asymptotically optimal statistical procedures can be carried over from one experiment to the other. The equivalence given here is established under a weak assumption on the parameter space $\mathcal{F}$. In particular, a sharp Besov smoothness condition is given on $\mathcal{F}$ which is sufficient for Poissonization, namely, if $\mathcal{F}$ is in a Besov ball $B_{p,q}^α(M)$ with $αp>1/2$. Exam…

Some thoughts on Le Cam's statistical decision theory

The paper contains some musings about the abstractions introduced by Lucien Le Cam into the asymptotic theory of statistical inference and decision theory. A short, self-contained proof of a key result (existence of randomizations via convergence in distribution of likelihood ratios), and an outline of a proof of a local asymptotic minimax theorem, are presented as an illustration of how Le Cam's approach leads to conceptual simplifications of asymptotic theory.

The Convolution Theorem of Hajek and Le Cam - Revisited

The present paper establishes convolution theorems for regular estimators when the limit experiment is non-Gaussian or of infnite dimension with sparse parameter space. Applications are given for Gaussian shift experiments of infnite dimension, the Brownian motion signal plus noise model, Levy processes which are observed at discrete times and estimators of the endpoints of densities with jumps. The method of proof is also of interest for the classical convolution theorem of Hajek and Le Cam. As technical tool we present an elementary approach for the comparison of limit experiments on standar…

浅层神经网络的定量泛函中心极限定理

A Quantitative Functional Central Limit Theorem for Shallow Neural Networks

本文针对单隐层神经网络建立了定量泛函中心极限定理，利用 Stein-Malliavin 泛函方法刻画了收敛速度对激活函数光滑性的依赖关系。