Local limit theorems for inhomogeneous Markov chains

An inhomogeneous Markov chain is a Markov chain $\{X_i\}$ whose set of states and transition kernels are allowed to vary in time. In his PhD, R. Dobrushin proved a central limit theorem for sums of the form

$$S_N=f_1(X_1,X_2)+···+f_N(X_N,X_{N+1})$$

I will discuss the local limit theorem in case $f_i$ are uniformly bounded and $\{X_i\}$ is uniformly elliptic.

This is joint work with D. Dolgopyat.