Two proofs of the KMT theorems

Abstract: The Komlós-Major-Tusnády theorem for simple symmetric random walk asserts that up to n steps, its path can be coupled to stay within distance log(n) of a Brownian motion run for time n. A second KMT theorem says that the empirical distribution function of n i.i.d. uniform random variables on [0,1] can be coupled to stay within log(n)/√n distance of a Brownian bridge.

Adding the idea of Cauchy criterion to existing proof architectures, we obtain (perhaps) simpler proofs of the above theorems. The first proof compares two Binomial distributions by combinatorial methods. The second proof compares Binomial and hypergeometric distributions among themselves by coupling Markov chains with these as stationary distributions. This is based on Chatterjee's proof via a form of Stein's method.

The first lecture will give an overview and the essence of the first proof. The second lecture will give an account of the second proof. Despite the statement of the main results, much of the lecture should be accessible (without knowing about Brownian motion) to those who know Markov chains.

probability

Audience: advanced learners

Bangalore Probability Seminar

Series comments: The link to zoom meeting can be found on the seminar's google calendar - www.isibang.ac.in/~d.yogesh/BPS.html