Limit Theorems and Some Applications in Statistical Physics
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Beratende Herausgeber/Advisory Editors:
Herbert Kurke, Berlin Joseph Mecke, Jena Rudiger Thiele, Leipzig Hans Triebel, Jena Gerd Wechsung, Jena
Ruben Ambartzumian, Jerevan David E. Edmunds, Brighton Alois Kufner, Prag Burkhard Monien, Paderborn Rolf J. Nessel, Aachen Claudio Procesi, Rom Kenji Ueno, Kyoto
Boris Nahapetian
Limit Theorems and Some Applications in Statistical Physics
B. G. Teubner Verlagsgesellschaft Stuttgart· Leipzig 1991
ISBN 978-3-322-93433-8 ISBN 978-3-322-93432-1 (eBook) DOI 10.1007/978-3-322-93432-1
Recent years have witnessed much progress in the theory of summation of dependent random variables. The book describes the contemporary state of several parts of this theory as well as its applications in the theory of Gibbs random fields, the basicals of which are also presented.
In den letzten Jahren wurden groBe Fortschritte in der Theorie der Summation abhangiger Variabler erzielt. Dieses Such beschreibt den gegenwartigen Stand sowohl der Theorie als auch ihrer Anwendungen in der Theorie der Gibbsschen zufalligen Felder. Au cours des dernieres annees, de grands progres ont ete realises dans la theorie de la sommation des variables eleatoires dependentes. Ce livre decrit l'etat actuel de certains chapitres de cette theorie, tout autant que ses applications a la theorie des champs aleatoires de Gibbs dont on presente les bases.
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PRE F ACE During the recent years the interest in limit theorems for dependent random variables has considerably grown being stimulated both by requirements of probability theory and by many applied disciplines. statistical A special place among such disciplines belongs to physics whose mathematical apparatus for a long time has been developing regardless the probability theory. However recently a process of mutual understanding between the specialists of this two fields started to develop. This resulted in a burst of activity for the benefit of both the subjects. For instance, the concept of Gibbs random field which has originated in statistical physics became a powerful stimulus for development of the theory of random fields while the Gibbsian fields themselves turned out to be excellent object for application of many general probabilistic results. The present book attempts to reflect this development concentrating upon limit theorems for sums of components of weakly dependent random processes and fields. In the first chapter we formulate some necessary facts from probability theory. The second chapter discusses different weak dependence conditions for random processes and fields. In Chapter 3 the behaviour of the variance of the sum of dependent random variables is investigated and some estimates for moments are given. The last part of this chapter presents
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