On weak convergence of functionals on smooth random functions

The numbers of level crossings and extremes for random processes and fields play an important role in reliability theory and many engineering applications. In many cases for Gaussian processes the Poisson approximation for their asymptotic distributions is used. This paper extends an approach proposed in Rusakov and Seleznjev (1988) for smooth random processes on a finite interval. It turns out that a number of functionals (including some integervalued ones) become continuous on the space of smooth functions and weak convergence results for the sequences of such continuous functionals are applicable. Examples of such functionals for smooth random processes on infinite intervals and for random fields are studied

Boundary non-crossing probabilities for fractional Brownian motion with trend

In this paper, we investigate the boundary non-crossing probabilities of a fractional Brownian motion considering some general deterministic trend function. We derive bounds for non-crossing probabilities and discuss the case of a large trend function. As a by-product, we solve a minimization problem related to the norm of the trend function

Approximation of a random process with variable smoothness

This Festschrift in honour of Paul Deheuvels’ 65th birthday compiles recent research results in the area between mathematical statistics and probability theory with a special emphasis on limit theorems. The book brings together contributions from invited international experts to provide an up-to-date survey of the field. Written in textbook style, this collection of original material addresses researchers, PhD and advanced Master students with a solid grasp of mathematical statistics and probability theory

On scrimshaw precursors: a 13th-century carved and engraved sperm whale tooth

In der einschlägigen Forschung zur Walfängervolkskunst des "Scrimshaw" - den seit den 1820er Jahren populär gewordenen Seemannsarbeiten speziell auf und aus Pottwal- und Walroßzahn, Barten und Walknochen - werden künstlerisch bearbeitete Gegenstände anderer Kulturen aus denselben Materialien als "Scrimshaw"-Vorläufer bezeichnet. Aus dem europäischen Mittelalter sind zwar zahlreiche Beispiele von Schnitzereien aus Walroßzahn und Walknochen bekannt, nicht jedoch aus Pottwalzahn, dem beinah stereotypen Werkstoff der "Scrimshander" des 19. und 20. Jahrhunderts. Der "Königsspiegel", ein um 1250/60 in Norwegen geschriebener pädagogischer Text, erwähnt gleichwohl Schnitzereien aus Pottwalzahn. Auch aus dem 16. und 17. Jahrhundert sind textliche Hinweise auf skandinavisches Kunstgewerbe aus den Zähnen dieses Meeressäugers bekannt. Im Anschluß an die Präsentation dieser Quellen wird ein konkretes Beispiel vorgestellt: Es handelt sich um ein beschnitztes und graviertes Salbenhorn aus Pottwalzahn, das sich in der Sammlung christlicher Kunst und Kultur des Museums der Universität Bergen, Norwegen, lnv. MA 437, befindet. Aus dem massiven Dentin wurde ein Greif herausgeschnitzt, der Ausguß wurde wie der Kopf eines gotischen Wasserspeiers gestaltet. Auf einer glatten Seitenfläche wurde später ein kleines Tondo mit Pflanzen- und Vogelmotiv eingraviert. Es trägt eine Inschrift, die vom Bergenser Museum als griechisch, hier aber als russisch identifiziert wurde, deren Sinn aber unklar bleibt. Anlass, die vom Museum vorgenommene Datierung des Zahns in das 13. Jahrhundert zu revidieren, besteht allerdings nicht

Asymptotic properties of keys and functional dependencies in random databases

AbstractPractical database applications give the impression that sets of constraints are rather small and that large sets are unusual and are caused by bad design decisions. Theoretical investigations, however, show that minimal constraint sets are potentially very large. Their size can be estimated to be exponential in terms of the number of attributes. The gap between observation in practice and theory results in the rejection of theoretical results. However, practice is related to average cases and is not related to worst cases.The theory used until now considered the worst-case complexity. This paper aims to develop a theory for the average-case complexity. Several probabilistic models and asymptotics of corresponding probabilities are investigated for random databases formed by independent random tuples with a common discrete distribution. Poisson approximations are studied for the distributions of some characteristics for such databases where the number of tuples is sufficiently large. We intend to prove that the exponential complexity of key sets and sets of functional dependencies is rather unusual and almost all minimal keys in a relation have a length which depends mainly on the size of the relation

The average length of keys and functional dependencies in (random) databases

. Practical database applications engender the impression that sets of constraints are rather small and that large sets are unusual and caused by bad design decisions. Theoretical investigations show, however, that minimal constraint sets are potentially very large. Their size can be estimated to be exponential in terms of the number of attributes. The gap between belief and theory causes non-acceptance of theoretical results. However, beliefs are related to average cases. The theory known so far considered worst case complexity. This paper aims at developing a theory of average case complexity. Several statistic models and asymptotics of corresponding probabilities are investigated for random databases. We show that exponential complexity of independent key sets and independent sets of functional dependencies is rather unusual. Depending on the size of relations almost all minimal keys have a length which mainly depends on the size. The number of minimal keys of other length is expone..