Reduction theory in the non-stationary setup

The central limit theorem (CLT) is one of the main results in probability theory. Its local version (LCLT) has origins in the De Moivre-Laplace theorem and it precedes it in that sense.

For partial sums of certain classes of stationary processes the LCLT is tied up with reducibility of stochastic processes to lattice valued random variables. By now results in the stationary case are classical and they rely on the behavior of the spectrum of the appropriate semigroup of operators.

For uniformly elliptic of inhomogeneous Markov chains the LCLT problem was “solved” recently by D. Dolgopyat and O. Sarig (partially motivated by problems in dynamics). In the talk we will discuss the corresponding theory for a wide class of non-stationary dynamical systems and discuss applications.

As will be discussed, the reducibility theory has an additional ingredient in the non-stationary setup.

Based on a joint work with Dmitry Dolgopyat.