Subgeometric ergodicity and $\beta$-mixing

Understand

It is well known that stationary geometrically ergodic Markov chains are $\beta$-mixing (absolutely regular) with geometrically decaying mixing coefficients.

• Furthermore, for initial distributions other than the stationary one, geometric ergodicity implies $\beta$-mixing under suitable moment assumptions.
• In this note we show that similar results hold also for subgeometrically ergodic Markov chains.
• In particular, for both stationary and other initial distributions, subgeometric ergodicity implies $\beta$-mixing with subgeometrically decaying mixing coefficients.