One idea and two proofs of the KMT theorems

Bounds on the running maximum of a random walk with small drift

“…The difficulty of the original KMT proof has motivated several recent attempts at simplification and better understanding of the result, such as Bhattacharjee and Goldstein [1], Chatterjee [3], and Krishnapur [8]. There is another strong approximation result due to Sakhanenko [9] which, according to p. 232 of [3], "is so complex that some researchers are hesitant to use it".…”