Uniform integrability condition in strong ration limit theorems

For a given Markov chain with a measurable state space $(E,\mathscr{E})$, transition operator $P$, and fixed measurable function $f\geq 0$, under necessary conditions, we consider variables $\mu(f_n)$, where $n\ge 1$ is sufficiently large, $f_n=P^nf/\nu(P^nf)$, and $\mu$ and $\nu$ are probability measures on $\mathscr{E}$. For a wide class of situations we propose sufficient and often necessary and sufficient conditions for the convergence of $f_n$ to 1 as $n\to\infty$. These results differ from the results of Orey, Lin, Nummelin, and others by replacing the traditional recurrent conditions of a chain or the uniform boundedness of the functions $f_n$ and the minorizing condition of [E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, Cambridge University Press, Cambridge, UK, 1984] with more flexible assumptions, among which the uniform integrability of functions $f_n$ with respect to some collection of measures plays a particular role. Our theorems imply a weak and often a strong convergence of these functions to $\varphi\equiv 1$ in respective spaces of a summable function.