Infinite-dimensional and continuous-time moving average processes

Abstract

This thesis consists of two parts both dealing with topics in time series analysis. In Chapter 2 we study necessary and sufficient conditions for the existence of strictly stationary solutions of ARMA equations in a separable complex Banach space. First, we obtain conditions for ARMA(1,q) equations by excluding zero and the unit circle from the spectrum of the operator of the AR part, where we use a decomposition similar to the Jordan decomposition of matrices. We then extend this to ARMA(p,q) equations by using a state space representation of an ARMA(p,q) process as an ARMA(1,q) process. We also show that many ARMA processes in Banach spaces possess a moving average process representation where the coefficients can be calculated as the coefficients of a Laurent series. Finally, we discuss various examples illustrating what may happen if one drops the assumptions we made. In Chapter 3 we study the asymptotic behaviour of the covariance estimator for a continuous-time moving average process with long memory. We choose the kernel function to be decaying polynomially slowly at infinity such that the continuous-time moving average process exhibits the long-memory property. We then show, depending on the speed of the polynomial decay of the kernel function and on the tail behaviour of the driving Levy process, that the covariance estimator is asymptotically Rosenblatt, stable or normal distributed.