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Description:
This thesis covers miscellaneous topics in the field of time series analysis and stochastic processes and consists of four topics where the first two are connected by the appearance of random coefficients and the last two by inference of Lévy driven continuous time moving average processes. In Chapter 2, we consider a random recurrence equation driven by a bivariate i.i.d. sequence. Much attention has been paid to causal strictly stationary solutions of that random recurrence equation, i.e. to strictly stationary solutions of this equation when the start variable is assumed to be independent of the driving sequence. For this case, a complete characterization when such causal solutions exist can be found in literature. We shall dispose of the independence assumption and derive necessary and sufficient conditions for a strictly stationary, not necessarily causal solution of this equation to exist. In Chapter 3, we introduce a continuous time autoregressive moving average process with random Lévy coefficients, termed RC-CARMA(p,q) process, of order p and q < p via a subclass of multivariate generalized Ornstein-Uhlenbeck processes. Sufficient conditions for the existence of a strictly stationary solution and the existence of moments are obtained. We further investigate second order stationarity properties, calculate the auto- covariance function and spectral density, and give sufficient conditions for their existence. In Chapter 4, we study a Lévy driven continuous time moving average process X sampled at random times which follow a renewal structure independent of X. Asymptotic normality of the sample mean, the sample autocovariance, and the sample autocorrelation is established under certain conditions on the kernel and the random times. We compare our results to a classical non-random equidistant sampling method and give an application to parameter estimation of the Lévy driven Ornstein-Uhlenbeck process. As an extension of the results in Chapter 4, we consider in Chapter 5 multivariate Lévy driven continuous ...