دانلود کتاب Probability Theory

"This book is intended to serve as a guide to the student of probability theory.» That sentence stands at the beginning of the Preface to my book Probability Theory and Elements of MelJS1Jf'e Theory, which was published in 1972 (by Holt, Rinehart and Winston, Inc., New York) and in 1981 in a second edition (by Academic Press Inc., London, Ltd.). Now 23 years later a new book is appearing, bearing a new title: Probability Theory. The question naturally arises whether and to what extent the contents and the aims of this new book have changed. The second part of this question is easier to answer: The aim remains the same, to serve as a reliable guide to those studying probabiliy theory. The answer to the first part has several components: On one hand, the first part of the old book devoted to measure theory has been eliminated. Actually it has been extensively r&worked and was published in German (first edition 1990, second 1992), under the title Mo.ss- und Integmtionstheorie. An English translation is in preparation. All this in response to a wish expressed by many earlier readers. The part of the older book devoted to probability theory has been extensively re-written. A new conception seemed n ecessary in order to better orient the book toward contemporary developments. An introductury text can no longer claim to bring the reader to the absolute frontiers of research. The pace of the latter in probability theory has been far too rapid in the last two decades. A book like this must, however, open up for the reader the possibility of progressing further with minimal strain into the specialized literature. I kept this requirement constantly in mind while writing the book. The idea of a guide is also to be understood in this sense: The reader should be led to hike through basic terrain along well-secured paths, now and then even scrambling up to a particular prominence in order to get an overview of a region. After this he should be prepared to forge ahead into less-developed parts of the terrain, if need be with special guides or, if research drives him, to penetrate into wholly new territory on his own. Among the most significant features of the book, a few should be emphasized here: Commensurate with its importance, martingale theory is gone into quite early and deeply. The law of the iterated logarithm is proved in a form which goes back to V. Strassenj this considerably sharpens the classical theorem of Ph. Hartman and A. Wintner. Two long chapters are devoted to the theory of stochastic processes. In particular,. Brownian motion - nowadays a fundamental mathematical concept - and the Ornstein-Uhlenbeck process are discussed in great detail. Also the style of exposition has changed: Some redundency was consciously built in here and there. The decisive determinant of the value of a textbook is, however, the reliability and precision of what it promulgates. Reliability includes the demand that the text must be self-contained in the sense that proofs always be complete: The reader must not be referred to exercises in order to reduce the length of a proof. So the only prerequisite for rAALIing this book is a sufficient knowledge of measure and integration theory. Any standard textbook devoted to this subject will provide the reader with the n ecessary background and details. This book is much more than a pure translation of the German original (cf. BAUER (1991) in the Bibliography). It is in fact a revised and improved version of that book. A translator, in the strict sense of the word, could never do this job. This explains why I have to express my deep gratitude to my very special translator, to my American colleague Professor Robert B. Burckel from Kansas State University. He had gotten to know my book by reading its very first German edition. I owe our friendship to his early interest in it. He expended great energy, especially on this new book, using his extensive acquaintance with the literature to make many knowledgeable suggestions, pressing for greater clarity and giving intensive support in bringing this enterprise to a good conclusion. Contents:Preface Table of Contents Interdependence of chapters Notation Introduction Chapter I Basic Concepts or the Theory 1 Probability spaces and the language of probability theory 2 Laplace experiments and conditional probabilities 3 Random variables: Distribution, expected value, variance, Jensen's inequality 4 Special distributions and their properties 5 Convergence of random variables and distributions Chapter II Independence 6 Independent events and u-algebras 7 Independent random variablcs 8 Products and sums of independent random variables 9 Infinite products of probability spaces Chapter III Laws or Large Numbers 10 Posing the question 11 Zero-one laws 12 Strong Law of Large Numbers 13 Applications 14 Almost sure convergence of infinite series Chapter IV Martingales 15 Conditional expectations 16 Martingales - definition and examples 17 Transformation via optional times 18 Inequalities for supermartingales 19 Convergence theorems 20 Applications Chapter V Fourier Analysis 178 21 Integration of complex-valued functions 178 22 Fourier transformation and characteristic functions 181 23 Uniqueness and Continuity Theorems 190 24 Normal distribution and independence 203 25 Differentiability of Fourier transforms 208 26 Continuous mappings into the circle 215 Chapter VI Limit Distributions 219 27 Examples of limit theorems 219 28 The Central Limit Theorem 232 29 Infinitely divisible distributions 245 30 Gauss measures and multi-dimensional central limit theorem 256 Chapter VII Law of the Iterated Logarithm 266 31 Posing the question and elementary preparations 266 32 Probabilistic preparations 274 33 Strassen's theorem of the iterated logarithm 283 34 Supplements 291 Chapter VIII Construction of Stochastic Processes 297 35 Projective limits of probability measures 297 36 Kernels and semigroups of kernels 305 37 Processes with stationary and independent increments 319 38 Processes with pre-assigned path-set 327 39 Continuous modifications 333 40 Brownian motion as a stochastic process 340 41 Poisson processes 351 42 Markov processes 357 43 Gauss processes 372 44 Conditional distributions 387 Chapter IX Brownian Motion 395 45 Brownian motion with filtration and martingales 395 46 Maximal inequalities for martingales 401 47 Behavior of Brownian paths 407 48 Examples of stochastic integrals 420 49 Optional times and optional sampling 435 50 The strong Markov property 450 51 Prospectus 476 Bibliography 493 Symbol Index 501 Name Index 505 General Index 509