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We presented a proof for the classical stable limit laws under use of contraction method in combination with the Zolotarev metric. Furthermore, a stable limit law was proved for scaled sums of growing into sequences. This limit law was alternatively formulated for sequences of random variables defined by a simple degenerate recursion.
Für balancierte, irreduzible Pólya-Urnen-Modelle sind Grenzwertsätze für die normalisierte Anzahl von Kugeln einer Farbe bekannt. Für eine spezielle Urne, deren Dynamik mit "Randomised-Play-the-Winner Rule" bezeichnet wird, werden im Rahmen der bekannten Grenzwertsätze Konvergenzraten in Wasserstein-Metriken und in der Kolmogorov-Metrik im Falle eines nicht-normalverteilten Grenzwerts hergeleitet.
Der Hoppe-Baum ist eine zufällig wachsende, diskrete Baumstuktur, wobei die stochastische Dynamik durch die Entwicklung der Hoppe Urne wie folgt gegeben ist: Die ausgezeichnete Kugel mit der die Hoppe Urne startet entspricht der Wurzel des Hoppe Baumes. In der Hoppe Urne wird diese Kugel mit Wahrscheinlichkeit proportional zu einem Parameter theta>0 gezogen, alle anderen Kugeln werden mit Wahrscheinlichkeit proportional zu 1 gezogen. Wann immer eine Kugel gezogen wird, wird sie zusammen mit einer neuen Kugel in die Urne zurückgelegt, was in unserem Baum dem Einfügen eines neuen Kindes an den gezogenen Knoten entspricht. Im Spezialfall theta=1 erhält man einen zufälligen rekursiven Baum.
In der Arbeit werden Erwartungswerte, Varianzen und Grenzwertsätze für Tiefe, Höhe, Pfadlänge und die Anzahl der Blätter gegeben.
Im Rahmen dieser Arbeit wird der aktuelle Stand auf dem Gebiet des Lokalen Lovász Lemmas (LLL) beschrieben und ein Überblick über die Arbeiten zu konstruktiven Beweisen und Anwendungen gegeben. Ausgehend von Jószef Becks Arbeit zu einer algorithmischen Herangehensweise, haben sich in den letzten Jahren im Umfeld von Moser und Tardos und ihren Arbeiten zu einem konstruktiven Beweis des LLL eine erneute starke Beschäftigung mit dem Thema und eine Fülle von Verbesserungen entwickelt.
In Kapitel 1 wird als Motivation eine kurze Einführung in die probabilistische Methode gegeben. Mit der First- und Second Moment Method werden zwei einfache Vorgehensweisen vorgestellt, die die Grundidee dieses Beweisprinzips klar werden lassen. Von Paul Erdős eröffnet, beschreibt es Wege, Existenzbeweise in nicht-stochastischen Teilgebieten der Mathematik mithilfe stochastischer Überlegungen zu führen. Das Lokale Lemma als eine solche Überlegung entstammt dieser Idee.
In Kapitel 2 werden verschiedene Formen des LLL vorgestellt und bewiesen, außerdem wird anhand einiger Anwendungsbeispiele die Vorgehensweise bei der Verwendung des LLL veranschaulicht.
In Kapitel 3 werden algorithmische Herangehensweisen beschrieben, die geeignet sind, von der (mithilfe des LLL gezeigten) Existenz gewisser Objekte zur tatsächlichen Konstruktion derselben zu gelangen.
In Kapitel 4 wird anhand von Beispielen aus dem reichen Schatz neuerer Veröffentlichungen gezeigt, welche Bewegung nach der Arbeit von Moser und Tardos entstanden ist. Dabei beleuchtet die Arbeit nicht nur einen anwendungsorientierten Beitrag von Haeupler, Saha und Srinivasan, sondern auch einen Beitrag Terence Taos, der die Beweistechnik Mosers aus einem anderen Blickwinkel beleuchtet.
In this thesis, the asymptotic behaviour of Pólya urn models is analyzed, using an approach based on the contraction method. For this, a combinatorial discrete time embedding of the evolution of the composition of the urn into random rooted trees is used. The recursive structure of the trees is used to study the asymptotic behavior using ideas from the contraction method.
The approach is applied to a couple of concrete Pólya urns that lead to limit laws with normal distributions, with non-normal limit distributions, or with asymptotic periodic distributional behavior.
Finally, an approach more in the spirit of earlier applications of the contraction method is discussed for one of the examples. A general transfer theorem of the contraction method is extended to cover this example, leading to conditions on the coefficients of the recursion that are not only weaker but also in general easier to check.
This thesis covers the analysis of radix sort, radix select and the path length of digital trees under a stochastic input assumption known as the Markov model.
The main results are asymptotic expansions of mean and variance as well as a central limit theorem for the complexity of radix sort and the path length of tries, PATRICIA tries and digital search trees.
Concerning radix select, a variety of different models for ranks are discussed including a law of large numbers for the worst case behavior, a limit theorem for the grand averages model and the first order asymptotic of the average complexity in the quantile model.
Some of the results are achieved by moment transfer techniques, the limit laws are based on a novel use of the contraction method suited for systems of stochastic recurrences.
Urn models are simple examples for random growth processes that involve various competing types. In the study of these schemes, one is generally interested in the impact of the specific form of interaction on the allocation of elements to the types. Depending on their reciprocal action, effects of cancellation and self-reinforcement become apparent in the long run of the system. For some urn models, the influencing is of a smoothing nature and the asymptotic allocation to the types is close to being a result of independent and identically distributed growth events. On the contrary, for others, almost sure random tendencies or logarithmically periodic terms emerge in the second growth order. The present thesis is devoted to the derivation of central limit theorems in the latter case. For urns of this kind, we use a "non-classical" normalisation to derive asymptotic joint normality of the types. This normalisation takes random tendencies and phases into account and consequently involves random centering and, also, possibly random scaling.
Within the last thirty years, the contraction method has become an important tool for the distributional analysis of random recursive structures. While it was mainly developed to show weak convergence, the contraction approach can additionally be used to obtain bounds on the rate of convergence in an appropriate metric. Based on ideas of the contraction method, we develop a general framework to bound rates of convergence for sequences of random variables as they mainly arise in the analysis of random trees and divide-and-conquer algorithms. The rates of convergence are bounded in the Zolotarev distances. In essence, we present three different versions of convergence theorems: a general version, an improved version for normal limit laws (providing significantly better bounds in some examples with normal limits) and a third version with a relaxed independence condition. Moreover, concrete applications are given which include parameters of random trees, quantities of stochastic geometry as well as complexity measures of recursive algorithms under either a random input or some randomization within the algorithm.
Within the last twenty years, the contraction method has turned out to be a fruitful approach to distributional convergence of sequences of random variables which obey additive recurrences. It was mainly invented for applications in the real-valued framework; however, in recent years, more complex state spaces such as Hilbert spaces have been under consideration. Based upon the family of Zolotarev metrics which were introduced in the late seventies, we develop the method in the context of Banach spaces and work it out in detail in the case of continuous resp. cadlag functions on the unit interval. We formulate sufficient conditions for both the sequence under consideration and its possible limit which satisfies a stochastic fixed-point equation, that allow to deduce functional limit theorems in applications. As a first application we present a new and considerably short proof of the classical invariance principle due to Donsker. It is based on a recursive decomposition. Moreover, we apply the method in the analysis of the complexity of partial match queries in two-dimensional search trees such as quadtrees and 2-d trees. These important data structures have been under heavy investigation since their invention in the seventies. Our results give answers to problems that have been left open in the pioneering work of Flajolet et al. in the eighties and nineties. We expect that the functional contraction method will significantly contribute to solutions for similar problems involving additive recursions in the following years.
