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Recent developments in empirical processes

procedure. It is most useful for modelling discrete-valued variables. We discuss stationarity and mixing properties of these series, and properties of lengths of runs of given values. The runs properties are useful for fitting the models to data. For the purely autoregressive model, DAR(p), which is a particular pth-order Markov chain, initial conditions for stationarity of the process in the cases p = 1, 2, 3 are given. Thus it is possible to generate stationary sequences of discrete random variables with correlation structure specified by p parameters. Applications of the models to the analysis of epidemiological data, rainfall data and discrete-time point processes are being considered.