Mean convergence of Markovian spherical averages for measure-preserving actions of the free group

Mean convergence of Markovian spherical averages is established for a measure-preserving action of a finitely-generated free group on a probability space. We endow the set of generators with a generalized Markov chain and establish the mean convergence of resulting spherical averages in this case under mild nondegeneracy assumptions on the stochastic matrix $\Pi$ defining our Markov chain. Equivalently, we establish the triviality of the tail sigma-algebra of the corresponding Markov operator. This convergence was previously known only for symmetric Markov chains, while the conditions ensuring convergence in our paper are inequalities rather than equalities, so mean convergence of spherical averages is established for a much larger class of Markov chains.