  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. Although this result is
simple it should prove very useful in obtaining rates of mixing in situations
where geometric ergodicity can not be established. To illustrate our results we
derive new subgeometric ergodicity and $\beta$-mixing results for the
self-exciting threshold autoregressive model.
