INTEGRATION OF COVARIANCE KERNELS AND STATIONARITY

The necessary and sufficient matrix condition of Mitchell, Morris and Ylvisaker (1990) for a stationary Gaussian process to have a specified process as kth derivative is investigated. The mean-square smoothing approach of stationary processes requires integration of covariance functions preserving stationarity. By providing a recursive representation of the involved reproducing kernel Hilbert spaces it is possible to analyse another criterion for k-fold integration of a process. This criterion only contains inequalities for the variances of the integrated processes. If the Hilbert space associated with the covariance function has a special form, which often occurs, then it can be shown that such processes can be integrated arbitrarily often. This is especially the case for the Ornstein-Uhlenbeck process. The results are applied to the linear and the exponential kernel and yield explicit norms in the corresponding reproducing kernel Hilbert spaces for each integration.