On the autopersistence functions and the autopersistence graphs of binary autoregressive time series

The classical autocorrelation function may not be an effective and informative means in revealing the dependence features of a binary time series {yt}. Recently, the autopersistence functions defined as APF0(k) = P(yt+k = 1 | yt = 0) and APF1(k) = P(yt+k = 1 | yt = 1), k = 1, 2,…, have been proposed as alternatives to the autocorrelation function for binary time series. In this article we consider the theoretical autopersistence functions and their natural sample analogues, the autopersistence graphs, under a binary autoregressive model framework. Some properties of the autopersistence functions and the asymptotic properties of the autopersistence graphs are discussed. The results have potential application in the modelling of binary time series.