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 cannot be established. To illustrate our results we derive new subgeometric ergodicity and $\beta$ -mixing results for the self-exciting threshold autoregressive model.

Structural instability in economic time series is widely reported in the literature. It is most prevalent in such series as price indices and inflation related data. Many methods have been developed for analysing and modelling structural changes in a univariate time series model. However, most of them assume that the data are generated by one fixed type (linear or non-linear) of the time series processes. This paper proposes a strategy for modelling different segments of an economic time series by different linear or non-linear models. A graphical procedure is suggested for detecting the model change points. The proposed procedure is illustrated by modelling annual United Kingdom price inflation series over the period 1265 to 2005. Stochastic modelling of inflation rates is an important topic to actuaries for dealing with long-term index linked insurance business. The proposed method suggests dividing the U.K. inflation series into four segments for modelling. Inflation projections based on the latest segment of the data are obtained through simulations. To get a better understanding of the impact of structural changes on inflation projections we also perform a forecasting study.