论文标题
KMT定理的一个想法和两个证明
One idea and two proofs of the KMT theorems
论文作者
论文摘要
给出了嵌入定理的Komlós-Major-Tusnády的两个证据，一个用于统一的经验过程，一个用于简单的对称随机步行。更确切地说，证明是证明所需的单变量耦合结果，例如Tusnády的引理。这些证明是对现有证明体系结构的修改，一种组合（由Csörg®，Révész，Bretagnolle，Massart，Dudley，Carter，Pollard等）和一个分析（由于Sourav Chatterjee）。这两个证明都有一个共同的想法：我们比较彼此之间的二项式和高几何分布，而不是与高斯分布进行比较。在组合方法中，这涉及将二项式（N，1/2）分布与二项式（4N，1/2）分布进行比较，这主要涉及相应的二项式系数之间的比较。在分析方法中，这减少了Chatterjee的方法，使其在整数上耦合最近的邻居马尔可夫链，以使它们保持近距离。
Two proofs of the Komlós-Major-Tusnády embedding theorems, one for the uniform empirical process and one for the simple symmetric random walk, are given. More precisely, what are proved are the univariate coupling results needed in the proofs, such as Tusnády's lemma. These proofs are modifications of existing proof architectures, one combinatorial (the original proof with many modifications, due to Csörgõ, Révész, Bretagnolle, Massart, Dudley, Carter, Pollard etc.) and one analytical (due to Sourav Chatterjee). There is one common idea to both proofs: we compare binomial and hypergeometric distributions among themselves, rather than with the Gaussian distribution. In the combinatorial approach, this involves comparing Binomial(n,1/2) distribution with the Binomial(4n,1/2) distribution, which mainly involves comparison between the corresponding binomial coefficients. In the analytical approach, this reduces Chatterjee's method to coupling nearest neighbour Markov chains on integers so that they stay close.
