Variance bounding markov chains, L₂-Uniform mean ergodicity and the clt

We prove that variance bounding Markov chains, as defined by Roberts and Rosenthal [31], are uniformly mean ergodic in L₂ of the invariant probability. For such chains, without any additional mixing, reversibility, or Harris recurrence assumptions, the central limit theorem and the invariance principle hold for every centered additive functional with finite variance. We also show that L₂-geometric ergodicity is equivalent to L ₂-uniform geometric ergodicity. We then specialize the results to random walks on compact Abelian groups, and construct a probability on the unit circle such that the random walk it generates is L₂-uniformly geometrically ergodic, but is not Harris recurrent.