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Persistence exponents via perturbation theory : autoregressive and moving average processes

Kettner, Marvin (2021):
Persistence exponents via perturbation theory : autoregressive and moving average processes. (Publisher's Version)
Darmstadt, Technische Universität Darmstadt,
DOI: 10.26083/tuprints-00017566,
[Ph.D. Thesis]

Abstract

In this thesis, the persistence problem in the context of Markov chains is studied. We are mainly concerned with processes where the persistence probability converges to zero at exponential speed and we are interested in the rate of decay, the so-called persistence exponent. For the main results, we use methods from perturbation theory. This approach is completely new in the field of persistence. For this reason, we provide a mostly self-contained presentation of the used theorems of perturbation theory. We show that the persistence exponent of an autoregressive process of order one can be expressed as a power series in the parameter of the autoregressive process. Additionally, we derive an iterative formula for the coefficients of this power series representation. For moving average processes of order one similar results as in the autoregressive case are derived.

Item Type: Ph.D. Thesis
Erschienen: 2021
Creators: Kettner, Marvin
Status: Publisher's Version
Title: Persistence exponents via perturbation theory : autoregressive and moving average processes
Language: English
Abstract:

In this thesis, the persistence problem in the context of Markov chains is studied. We are mainly concerned with processes where the persistence probability converges to zero at exponential speed and we are interested in the rate of decay, the so-called persistence exponent. For the main results, we use methods from perturbation theory. This approach is completely new in the field of persistence. For this reason, we provide a mostly self-contained presentation of the used theorems of perturbation theory. We show that the persistence exponent of an autoregressive process of order one can be expressed as a power series in the parameter of the autoregressive process. Additionally, we derive an iterative formula for the coefficients of this power series representation. For moving average processes of order one similar results as in the autoregressive case are derived.

Place of Publication: Darmstadt
Collation: iv, 63 Seiten
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Stochastik
Date Deposited: 03 Mar 2021 12:33
DOI: 10.26083/tuprints-00017566
Official URL: https://tuprints.ulb.tu-darmstadt.de/17566
URN: urn:nbn:de:tuda-tuprints-175661
Referees: Aurzada, Prof. Dr. Frank and Wachtel, Prof. Dr. Vitali
Refereed / Verteidigung / mdl. Prüfung: 14 January 2021
Alternative Abstract:
Alternative abstract Language

In dieser Dissertation wird das Persistenz-Problem im Kontext von Markovketten studiert. In der vorliegenden Arbeit betrachten wir überwiegend Prozesse, bei denen die Persistenz-Wahrscheinlichkeit exponentiell schnell gegen Null konvergiert. Dabei ist von zentraler Bedeutung, die Rate dieses Abfallverhaltens zu ermitteln, den sogenannten Persistenz-Exponenten. Für die Hauptresultate dieser Dissertation werden Methoden der Störungstheorie bedient. Diese Vorgehensweise ist in dem Gebiet der Persistenz-Wahrscheinlichkeiten neu. Deshalb beinhaltet die Dissertation eine überwiegend eigenständige Präsentation der benötigten Resultate der Störungstheorie. Für einen autoregressiven Prozess der Ordnung eins zeigen wir, dass der Persistenz-Exponent als Potenzreihe im Parameter des autoregressiven Prozesses dargestellt werden kann. Ferner leiten wir eine iterative Formel für die Berechnung der Koeffizienten dieser Potenzreihe her. Für Moving-Average-Prozesse der Ordnung eins beweisen wir analoge Resultate wie für den Fall eines autoregressiven Prozesses.

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