We show how basic work of Don Burkholder on iterated conditional expectations is intimately connected to a standard tool of scientific computing—Glauber dynamics (also known as the Gibbs sampler). We begin with von Neumann’s alternating projection theorem using an example of Burkholder’s. We then review Burkholder’s theorem. Finally, we introduce Glauber dynamics and show how Burkholder’s theorem can be harnessed to prove convergence. In the other direction, we show how classical convergence rates involving the angle between subspaces can be substantially refined in several cases.
"Stochastic alternating projections." Illinois J. Math. 54 (3) 963 - 979, Fall; 2010. https://doi.org/10.1215/ijm/1336568522