Properties of seasonal long memory processes

We consider the fractional ARIMA process with seasonality s , denoted by SARFIMA ( p , d , q ) × ( P , D , Q ) s , which describes time series with long memory periodical behavior at finite number of spectrum frequencies. We present the proof of several properties of these processes, such as the spectral density function expression and its behavior near the seasonal frequencies, the stationarity, the intermediate and long memory characteristics, the autocovariance function and its asymptotic expression. We also investigate the ergodicity and we present necessary and sufficient conditions for the causality and the invertibility properties of SARFIMA processes.