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It is well known that stationary geometrically ergodic Markov chains are$\beta$-mixing (absolutely regular) with geometrically decaying mixingcoefficients. Furthermore, for initial distributions other than the stationaryone, geometric ergodicity implies $\beta$-mixing under suitable momentassumptions. In this note we show that similar results hold also forsubgeometrically ergodic Markov chains. In particular, for both stationary andother initial distributions, subgeometric ergodicity implies $\beta$-mixingwith subgeometrically decaying mixing coefficients. Although this result issimple it should prove very useful in obtaining rates of mixing in situationswhere geometric ergodicity can not be established. To illustrate our results wederive new subgeometric ergodicity and $\beta$-mixing results for theself-exciting threshold autoregressive model.

Subgeometric ergodicity and $\beta$-mixing