One idea and two proofs of the KMT theorems.

Two proofs of the Koml\'os-Major-Tusn\'ady 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 Tusn\'{a}dy's lemma. These proofs are modifications of existing proof architectures, one combinatorial (the original proof with many modifications, due to Cs\"{o}rg\~o, R\'{e}v\'{e}sz, 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.