Augmented GARCH sequences: Dependence structure and asymptotics

The augmented GARCH model is a unification of numerous extensions of the popular and widely used ARCH process. It was introduced by Duan and besides ordinary (linear) GARCH processes, it contains exponential GARCH, power GARCH, threshold GARCH, asymmetric GARCH, etc. In this paper, we study the probabilistic structure of augmented $\mathrm {GARCH}(1,1)$ sequences and the asymptotic distribution of various functionals of the process occurring in problems of statistical inference. Instead of using the Markov structure of the model and implied mixing properties, we utilize independence properties of perturbed GARCH sequences to directly reduce their asymptotic behavior to the case of independent random variables. This method applies for a very large class of functionals and eliminates the fairly restrictive moment and smoothness conditions assumed in the earlier theory. In particular, we derive functional CLTs for powers of the augmented GARCH variables, derive the error rate in the CLT and obtain asymptotic results for their empirical processes under nearly optimal conditions.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ120 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Eleven Years of QCD at LEP

Studies of hadronic final states of $e^+e^-$ annihilations, observed at the Large Electron Positron Collider LEP at CERN, are reviewed. The topics included cover measurements of $\alpha_s$, hadronic event shapes and hadronisation studies, tests of asymptotic freedom and of the non-Abelian gauge structure of QCD, differences between quark and gluon jets, tests of power corrections and selected results of two-photon scattering processes. The improvements obtained at LEP are demonstrated by comparing to results from the pre-LEP era. This article consists of a reproduction of slides presented at the LEPFest in October 2000, supplemented by a short descriptive text and a list of relevant references.Comment: 10 pages of text plus reproduction of 27 transparencies presented at the LEPFest at CERN, October 2000. To be published in Eur. Phys. Jour (direct) C; a higher resolution version of the viewgraphs can be obtained from: http://www.mppmu.mpg.de/~bethke/LEPQCDtalk-higres.pd

αs\alpha_s at Zinnowitz 2004

A review of measurements of $\alpha_s$ is given, representing the status of April 2004. The results prove the energy dependence of $\alpha_s$ and are in excellent agreement with the expectations of Quantum Chromodynamics, QCD. Evolving all results to the rest energy of the $Z^0$ boson, the world average of $\alpha_s(M_Z)$ is determined from measurements which are based on QCD calculations in complete NNLO perturbation theory, giving $\alpha_s(M_Z) = 0.1182 \pm 0.0027 .$Comment: 8 pages, 4 figures, talk presented at the workshop Loops and Legs in Quantum Field Theory, April 2004, Zinnowitz (Germany

Media Philosophy— A Reasonable Programme?

It is beyond any doubt that media have an enormous impact on our media-culture societies. Media in?uence our perception and our knowledge, our memory as well as our emotions. They create public spheres and public opinions and give rise to media realities. Media shape our socialisation and our communality. They transform economy, politics, science, religion and law. “What we know about our society, even about our world we are living in, we know via the mass media.” (Luhmann 1996:9; my translation) Accordingly, “the media” have become a paramount subject of interdisciplinary discourses in the last decades all over the world. All these developments have become topics of scienti?c analyses as well as parts of media programmes. Since decades, various academic disciplines focused on an other-observation (“Fremdbeobachtung”) of the media from an external state, whereas the media increasingly tend to observe themselves as well as one another in order to transform this self-observation into parts of their respective programmes. The other-observation is carried out either by scholars of communication- and/or media theory or by philosophers; but whereas the former are organised in academic disciplines, no established discipline entitled “media philosophy” exists until today. Instead, the various approaches to philosophical analyses of media are heterogeneous and lack a solid theoretical basis as well as a disciplinary organisation. Some scholars even hold the view that media are not even within the province of philosophers. Some people deeply regret this deadlock regarding not only topics and discourses but also future jobs and positions for scholars of a discipline “media philosophy” to come. Others welcome this stalemate which gives room to creative solutions of thematic as well as of organisational matters. Let us have a short look at some of the foreseeable options. One of the actual media philosophical approaches concentrates its efforts on a reformulation of traditional philosophical topics in the framework of media ef?ciencies. The list of such topics is rather long and covers nearly all famous crucial subjects of philosophical discourses, reaching from reality, truth, culture, society, education or politics to time, space, emotion, subject or entertainment. This kind of rethinking or reformulating philosophical topics concentrates upon the question how—in the co-evolution of media systems and society—our daily experiences as well as our theoretical modellings of these topics have changed on the historical way from writing to the Internet

Scaling in thermal convection: A unifying theory

A systematic theory for the scaling of the Nusselt number $Nu$ and of the Reynolds number $Re$ in strong Rayleigh-Benard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large scale convection roll (``wind of turbulence'') and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number versus Prandtl number phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra smaller than 10^{11}) the leading terms are $Nu\sim Ra^{1/4}Pr^{1/8}$, $Re \sim Ra^{1/2} Pr^{-3/4}$ for $Pr < 1$ and $Nu\sim Ra^{1/4}Pr^{-1/12}$, $Re \sim Ra^{1/2} Pr^{-5/6}$ for $Pr > 1$. In most measurements these laws are modified by additive corrections from the neighboring regimes so that the impression of a slightly larger (effective) Nu vs Ra scaling exponent can arise. -- The presented theory is best summarized in the phase diagram figure 1.Comment: 30 pages, latex, 7 figures, under review at Journal of Fluid Mec