Local limit theorem for inhomogeneous Markov chains (joint with Dolgopyat)

An inhomogeneous Markov chain $X_n$ is a Markov chain whose state spaces and transition kernels change in time. A “local limit theorem” is an asymptotic formula for probabilities of the form

$Prob[S_N-z_N\in (a,b)]$, $S_N=f_1(X_1,X_2)+....+f_N(X_N,X_{N+1})$

in the limit $N\to\infty$. Here $z_N$ is a “suitable” sequence of numbers. I will describe general sufficient conditions for such results.

If time allows, I will explain why such results are needed for the study of certain problems related to irrational rotations.

This is joint work with Dmitry Dolgopyat.