On the regularity of spectral densities of continuous-time completely linearly regular processes

This paper deals with the study of the relationship between the complete linear regularity of continuous-time weakly stationary processes and the smoothness of their spectral densities. It is shown that when the coefficient of complete linear regularity behaves like O([tau]-(r+[mu])) as [tau] --> +[infinity], for some , [mu] [set membership, variant] (0,1], then the spectral density has at least r uniformly continuous, bounded, and integrable derivatives, with the rth derivative satisfying a Lipschitz continuity condition of order [mu]. Conversely, under certain smoothness assumptions on the spectral density, upper bounds on the rate of decay of the coefficient of complete linear regularity are obtained.Weakly stationary stochastic process Continuous-time process Complete linear regularity Spectral density Strong mixing