概率論教程

前言

In this new edition, I have added a Supplement on Measure and Integral. The subject matter is first treated in a general setting pertinent to an abstract measure space, and then specified in the classic Borel-Lebesgue case for the real line. The latter material, an essential part of real analysis, is presupposed in the original edition published in 1968 and revised in the second edition of 1974. When I taught the course under the title "Advanced Probability" at Stanford University beginning in 1962, students from the departments of statistics, operations research （formerly industrial engineering）, electrical engi- neering, etc. often had to take a prerequisite course given by other instructors before they enlisted in my course. In later years I prepared a set of notes, lithographed and distributed in the class, to meet the need. This forms the basis of the present Supplement. It is hoped that the result may as well serve in an introductory mode, perhaps also independently for a short course in the stated topics.The presentation is largely self-contained with only a few particular refer- ences to the main text. For instance, after （the old） ~2.1 where the basic notions of set theory are explained, the reader can proceed to the first two sections of the Supplement for a full treatment of the construction and completion of a general measure; the next two sections contain a full treatment of the mathe- matical expectation as an integral, of which the properties are recapitulated in 3.2. In the final section, application of the new integral to the older Riemann integral in calculus is described and illustrated with some famous examples. Throughout the exposition, a few side remarks.

內(nèi)容概要

隨機(jī)變量和分布函數(shù)，測度論，數(shù)學(xué)期望，方差，各種收斂性，大數(shù)律， 中心極限定理，特征函數(shù)，隨機(jī)游動， 馬氏性和鞅理論.本書內(nèi)容豐富，邏輯緊密，敘述嚴(yán)謹(jǐn)，不僅可以擴(kuò)展讀者的視野，而且還將為其后續(xù)的學(xué)習(xí)和研究打下堅實基礎(chǔ)。此外，本書的習(xí)題較多， 都經(jīng)過細(xì)心的遴選， 從易到難， 便于讀者鞏固練習(xí)。本版補(bǔ)充了有關(guān)測度和積分方面的內(nèi)容，并增加了一些習(xí)題?！　”緯且槐鞠碜u(yù)世界的經(jīng)典概率論教材，令眾多讀者受益無窮，自出版以來，已被世界75%以上的大學(xué)的數(shù)萬名學(xué)生使用。本書內(nèi)容豐富，邏輯清晰，敘述嚴(yán)謹(jǐn)，不僅可以拓展讀者的視野，而且還將為其后續(xù)的學(xué)習(xí)和研究打下堅實基礎(chǔ)。此外，本書的習(xí)題較多, 都經(jīng)過細(xì)心的遴選, 從易到難, 便于讀者鞏固練習(xí)。本版補(bǔ)充了有關(guān)測度和積分方面的內(nèi)容，并增加了一些習(xí)題。

作者簡介

Kai Lai Chung（鐘開萊，1917-2009）華裔數(shù)學(xué)家、概率學(xué)家。浙江杭州人。1917年生于上海。1936年考入清華大學(xué)物理系。1940年畢業(yè)于西南聯(lián)合大學(xué)數(shù)學(xué)系，之后任西南聯(lián)合大學(xué)數(shù)學(xué)系助教。1944年考取第六屆庚子賠款公費(fèi)留美獎學(xué)金。1945年底赴美國留學(xué)。1947年獲普林斯頓大學(xué)博士學(xué)位。20世紀(jì)50年代任教于美國紐約州Syracuse大學(xué)，60年代以后任斯坦福大學(xué)數(shù)學(xué)系教授、系主任、名譽(yù)教授。鐘開萊著有十余部專著。為世界公認(rèn)的20世紀(jì)后半葉“概率學(xué)界學(xué)術(shù)教父”。

書籍目錄

Preface to the third edition Preface to the second edition Preface to the first edition 1 Distribution function 　1.1 Monotone functions 　1.2 Distribution functions 　1.3 Absolutely continuous and singular distributions 2 Measure theory 　2.1 Classes of sets 　2.2 Probability measures and their distribution functions 3 Random variable. Expectation. Independence 　3.1 General definitions 　3.2 Properties of mathematical expectation 　3.3 Independence 4 Convergence concepts 　4.1 Various modes of convergence 　4.2 Almost sure convergence; Borel-Cantelli lemma 　4.3 Vague convergence 　4.4 Continuation 　4.5 Uniform integrability; convergence of moments 5 Law of large numbers. Random series 　5.1 Simple limit theorems 　5.2 Weak law of large numbers 　5.3 Convergence of series 　5.4 Strong law of large numbers 　5.5 Applications 　Bibliographical Note 6 Characteristic function 　6.1 General properties; convolutions 　6.2 Uniqueness and inversion 　6.3 Convergence theorems 　6.4 Simple applications 　6.5 Representation theorems 　6.6 Multidimensional case; Laplace transforms 　　Bibliographical Note 7 Central limit theorem and its ramifications 　7.1 Liapounov's theorem 　7.2 Lindeberg-Feller theorem 　7.3 Ramifications of the central limit theorem 　7.4 Error estimation 　7.5 Law of the iterated logarithm 　7.6 Infinite divisibility 　Bibliographical Note 8 Random walk 　8.1 Zero-or-one laws 　8.2 Basic notions 　8.3 Recurrence 　8.4 Fine structure 　8.5 Continuation 　Bibliographical Note 9 Conditioning. Markov property. Martingale Bibliographical Note Supplement: Measure and Integral General Bibliography Index

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