  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.
