A THEORY OF CORRELATION DIMENSION FOR STATIONARY TIME-SERIES

We develop a formal theory of correlation dimension for a class of stationary time series that includes both deterministic outputs and gaussian processes with continuous paths. This theory enables us to completely analyse correlation dimension in gaussian processes via spectral methods. Our approach plus recent results on the convergence behaviour of the sample correlation integral are then used to re-examine the behaviour of gaussian power-law coloured noise. We show that the finite correlation dimension observed by Osborne and Provenzale (1989) is a local quantity entirely due to the non-ergodicity of the simulation model. We also show that non-ergodic and weakly ergodic finite-dimensional dynamical systems (such as the simple circle map) exhibit the same phenomenon.