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(τ−(r+μ)) as τ → +∞, for some r ∈ N, μ ∈ (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 μ. 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.