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Description:
In this thesis, we investigate long range dependence (LRD) of infinite-variance, stationary stochastic processes X = (Xt)t∈R. As many definitions of LRD are not applicable for infinite-variance processes, we make use of a recent definition of LRD in terms of the integrability of the covariance of indicators of excursion sets. This means that we consider covariances of the form Cov (1{X0 > u}, 1{Xt > v}) for time points t ∈ R and levels u, v ∈ R. The advantage of this approach is that the boundedness of the indicator function guarantees that this covariance is well-defined even if X has an infinite ariance. To distinguish this notion of LRD from other existing definitions of LRD, we will denote this as IE-LRD and the corresponding short range dependence as IE-SRD. We apply this new definition on selected infinite-variance process like symmetric α-stable moving averages, max-stable processes and subordinated Gaussian processes. Specifically, we derive conditions under which these processes are IE-LRD or IE-SRD. These conditions will be stated in terms of natural characteristics of the process like e.g. an α-stable moving average’s kernel function or a max-stable process’ extremal coefficients. Furthermore, we show that IE-LRD can be detected empirically. To do so, we consider a well-known estimator of the so-called memory parameter and show that it is consistent even if the process that the estimator is applied on is IE-LRD. So far, this estimator has only been used in finite-variance settings. Thus, the novelty of our results is not that the estimator is new but that consistency is proved in such a general setting so that IE-LRD can be detected empirically even for infinite-variance processes. Finally, we estimate a widely used dependence measure of max-stable time series, namely its tail dependence coefficients. Specifically, we show that a simple estimator of these coefficients is asymptotically normal when the time series is IE-SRD. Typically, these type of limit theorems assume complicated mixing ...