Course in Probability Theory

xiv I PREFACE TO THE FIRST EDITIONbest seen in the advanced study of stochastic processes, but will already beabundantly clear from the contents of a general introduction such as this book .Although many notions of probability theory arise from concrete modelsin applied sciences, recalling such familiar objects as coins and dice, genesand particles, a basic mathematical text (as this pretends to be) can no longerindulge in diverse applications, just as nowadays a course in real variablescannot delve into the vibrations of strings or the conduction of heat . Incidentally,merely borrowing the jargon from another branch of science withouttreating its genuine problems does not aid in the understanding of concepts orthe mastery of techniques .A final disclaimer : this book is not the prelude to something else and doesnot lead down a strait and righteous path to any unique fundamental goal .Fortunately nothing in the theory deserves such single-minded devotion, asapparently happens in certain other fields of mathematics . Quite the contrary,a basic course in probability should offer a broad perspective of the open fieldand prepare the student for various further possibilities of study and research .To this aim he must acquire knowledge of ideas and practice in methods, anddwell with them long and deeply enough to reap the benefits .A brief description will now be given of the nine chapters, with somesuggestions for reading and instruction . Chapters 1 and 2 are preparatory . Asynopsis of the requisite "measure and integration" is given in Chapter 2,together with certain supplements essential to probability theory . Chapter 1 isreally a review of elementary real variables ; although it is somewhat expendable,a reader with adequate background should be able to cover it swiftly andconfidently-with something gained from the effort . For class instruction itmay be advisable to begin the course with Chapter 2 and fill in from Chapter 1as the occasions arise . Chapter 3 is the true introduction to the language andframework of probability theory, but I have restricted its content to what iscrucial and feasible at this stage, relegating certain important extensions, suchas shifting and conditioning, to Chapters 8 and 9 . This is done to avoid overloadingthe chapter with definitions and generalities that would be meaninglesswithout frequent application . Chapter 4 may be regarded as an assembly ofnotions and techniques of real function theory adapted to the usage of probability. Thus, Chapter 5 is the first place where the reader encounters bona fidetheorems in the field . The famous landmarks shown there serve also to introducethe ways and means peculiar to the subject . Chapter 6 develops some ofthe chief analytical weapons, namely Fourier and Laplace transforms, neededfor challenges old and new . Quick testing grounds are provided, but for majorbattlefields one must await Chapters 7 and 8 . Chapter 7 initiates what has beencalled the "central problem" of classical probability theory . Time has marchedon and the center of the stage has shifted, but this topic remains withoutdoubt a crowning achievement . In Chapters 8 and 9 two different aspects of