Refine
Year of publication
- 2012 (1)
Document Type
- Doctoral Thesis (1)
Language
- English (1)
Has Fulltext
- yes (1)
Is part of the Bibliography
- no (1)
Keywords
- Banach spaces (1)
- Donkers theorem (1)
- Zolotarev metric (1)
- analysis of algorithms (1)
- contraction method (1)
- functional limit theorems (1)
- partial match queries (1)
- searchtrees (1)
Institute
- Mathematik (1)
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.