Metrical Theory of Continued Fractions 2002 ed. [Kietas viršelis]

Collects current knowledge of the metrical theory of the regular continued fraction expansion and related representations of real numbers. Iosifescu (Romanian Academy) and Kraaikamp (Delft University of Technology) review the basic properties of the continued fraction expansion, then generalize Gauss' problem by an elementary approach as well as functional-theoretic methods. The second half of the book develops limit theorems for incomplete quotients and related random variables, and the ergodic properties of the regular continued fraction expansion, leading to strong laws of large numbers. Three appendices present some results from measure theory, regularly varying functions, and limit theorems for mixing random variables. Annotation (c) Book News, Inc., Portland, OR (booknews.com)

The book is essentially based on recent work of the authors. In order to unify and generalize the results obtained so far, new concepts have been introduced, e.g., an infinite order chain representation of the continued fraction expansion of irrationals, the conditional measures associated with, and the extended random variables corresponding to that representation. Also, such procedures as singularization and insertion allow to obtain most of the continued fraction expansions related to the regular continued fraction expansion.The authors present and prove with full details for the first time in book form, the most recent developments in solving the celebrated 1812 Gauss' problem which originated the metrical theory of continued fractions. At the same time, they study exhaustively the Perron-Frobenius operator, which is of basic importance in this theory, on various Banach spaces including that of functions of bounded variation on the unit interval.The book is of interest to research workers and advanced Ph.D. students in probability theory, stochastic processes and number theory.