Modelling Long‐memory Time Series with Finite or Infinite Variance: a General Approach

Abstract

We present a class of generalized fractional filters which is stable with respect to series and parallel connection. This class extends the so‐called fractional ARUMA and fractional ARMA filters previously introduced by e.g. Goncalves (1987) and Robinson (1994) and recently studied by Giraitis and Leipus (1995) and Viano et al. (1995). Conditions for the existence of the induced stationary SαS and L2 processes are given. We describe the asymptotic dependence structure of these processes via the codifference and the covariance sequences respectively. In the L2 case, we prove the weak convergence of the normalized partial sums.