An M-dependent process X(n) on the integers, is a process for which every event concerning with X(-1),X(-2),... is independent from every event concerning with X(M),X(M+1),...
Such processes play an important role both as scaling limits of physical systems and as a tool in approximating other processes.
A question that has risen independently in several contexts is:
"is there an M dependent proper colouring of the integer lattice for some finite M?"
Abstract: It was noticed in the 30's by Doeblin & Forte that Markov
operators with "chains with complete connections"
act quasi-compactly on the Lipschitz functions. These are operators
like the transfer operators of certain expanding
C^2 interval maps (e.g. the square of Gauss map).
It is folklore that stochastic processes generated by smooth
observables under these maps satisfy many of the results
of "classical probability theory" (e.g. CLT, Chernoff inequality).
I'll try to explain some of this in a "lunchtime" mode.
Here is a title and abstract for the lunch seminar:
Rigidity sequences for weakly mixing transformations
Abstract: I will present a recent result of Bassam Fayad
and Jean-Paul Thouvenot that shows that any rigidty
sequence for an irrational rotation is also a rigidity
sequence for some weakly mixing transformation.