Ergodic Theoretical Approach To Investigate Memory Properties Of Heavy Tailed Processes
A class of infinitely divisible processes includes not only well-known L´ vy processes, e but also a wide variety of processes such as the Gaussian and the stable processes, moving averages driven by L´ vy processes (e.g., Ornstein-Uhlenbeck processes), and hare monizable processes. This dissertation focuses on the limit theorems for heavy tailed stationary infinitely divisible processes of a certain integral form. In the light of a recently developed, ergodic theoretical approach, an infinitely divisible process with integral representation can be decomposed into two processes; one with short range dependence and the other with long range dependence. In the language of ergodic theory, the former process is generated by a dissipative flow, while the latter one is generated by a conservative flow. If the underlying flow is dissipative, the process is known to be identical to moving averages. On the other hand, only few attempts have been made on the study of the processes generated by conservative flows, and this dissertation discusses three limit theorems for such processes. Specifically, we establish the functional central limit theorem, the limit theorem on the sample autocovariance, and the functional limit theorem on the partial maxima. Taking advantage of some ergodic theoretical notions, called pointwise dual ergodicity, the memory length in the process whose underlying flow is conservative can be quantified by a single parameter. It then turns out that the growth rates of partial sums, autocovariances, and partial maxima, together with the properties of their weak limits, all depend on not only heaviness of the marginal tail but also the memory length. In particular, the limiting process in the functional central limit theorem constitutes a new class of stable process. Similarly, a new class of Fr´ chet process can be derived e as weak limits for the normalized partial maxima. These new classes of weak limits exhibit dramatically different features that have never been observed in the limiting processes for moving averages. Subsequently, we also propose a new notion, called a tail measure, as an infinitedimensional object that can measure the dependence of extremes of stochastic processes or random fields with regularly varying tails. Focusing on stationary infinitely divisible processes of integral forms, we will investigate the connection between the ergodic theoretical properties of tail measures and those of the probability laws of the processes.
Resnick, Sidney Ira; Nussbaum, Michael
Ph. D., Operations Research
Doctor of Philosophy
dissertation or thesis