Statistica Sinica 17(2007), 15-41

FREQUENCY ANALYSIS OF CHAOTIC

INTERMITTENCY MAPS WITH SLOWLY DECAYING

CORRELATIONS

R. J. Bhansali and M. P. Holland

University of Liverpool and University of Exeter

Abstract: Established stochastic models for discrete-time long-memory processes are linear and Gaussian, and commonly require that the $d$th fractional-difference of the process has short memory $0<d\leq 0.5$; if $-0.5 < d < 0$, the process is said to have an intermediate memory. Chaotic intermittency maps provide alternative non-linear, non-Gaussian models for both classes of processes. An asymptotic expression for the rate at which the correlations of symmetric cusp map decay is developed, and the class of extended symmetric cusp maps is introduced. The small-frequency asymptotics of the polynomial, cusp and logarithmic maps are investigated, and it is shown that these maps can produce spectra with $d = 0.5$ on the one hand, $d = 0$ on the other, and yet the corresponding processes are stationary and have long-memory. Asymptotic expressions are derived for studying the bias of the small-frequency periodogram ordinates with these maps. Finite sample behaviour is examined in a simulation study.