One idea and two proofs of the KMT theorems
One idea and two proofs of the KMT theorems
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国民党定理的一个想法和两个证明
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发表时间：
2020
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通讯作者：
Manjunath Krishnapur
中科院分区：
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作者：
Manjunath Krishnapur
给出了一致经验过程和简单对称随机游动的Komlos-Major-Tusnady嵌入定理的两个证明。更确切地说，所证明的是证明所需的单变量耦合结果，如Tusnady引理。这些证明是现有证明架构的修改，一个组合（原始证明有许多修改，由于Csorgo，Revesz，Bretagnolle，Massart，达德利，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：我们比较二项式分布和超几何分布，而不是高斯分布。在组合方法中，这涉及比较二项（n，1/2）分布与二项（4 n，1/2）分布，这主要涉及相应的二项系数之间的比较。在分析方法中，这减少了查特吉的方法耦合整数上的最近邻马尔可夫链，使它们保持接近。
Two proofs of the Komlos-Major-Tusnady 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 Tusnady's lemma. These proofs are modifications of existing proof architectures, one combinatorial (the original proof with many modifications, due to Csorgo, Revesz, 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.