Limit Theorems and Some Applications in Statistical Physics

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. A special place among such disciplines belongs to statistical 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 several probabilistic inequalities which are the necessary technical tools in our proofs of limit theorems. Chapter 4 is devoted to description of the basic methods of proofs. In particular we present a modification of Bernstein's method. The fifth chapter contains limit theorems for the random processes satisfying the so-called mixing criterions ( ~,~, p, ~ - mixing): Various versions of the central limit theorem (c.l.th.), the invariance principle, the law of the iterated logarithm and rate of convergence in c.l.th. In Chapter 6 we introduce generalized mixing conditions and apply them in some versions of c.l.th. Several resalts from non-commutative probability theory are presented too. The seventh chapter is devoted to limit theorems for random fields. Here again the corresponding limit theorems are given.