Probability with MartingalesProbability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the ThreeSeries Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measuretheoretic results are proved in full in appendices, so that the book is completely selfcontained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital rôle. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction. 
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Review: Probability with Martingales
User Review  Fausto Saleri  GoodreadsInefficient, almost useless for any student. Notations are confuse, theorems are basically left to the reader; the choice behind the appendix is simply wrong, because you need to read it to go through ... Read full review
Review: Probability with Martingales
User Review  Fausto Saleri  GoodreadsInefficient, almost useless for any student. Notations are confuse, theorems are basically left to the reader; the choice behind the appendix is simply wrong, because you need to read it to go through ... Read full review
Contents
More about exercises In compiling Chapter E which consists exactly  4 
of nth generation Zn 0 3 Use of conditional expectations 0 4 Extinction  10 
Events  23 
First BorelCantelli Lemma BCl 2 8 Definitions liminfEnEnev  27 
Integration  49 
Introductory remarks 6 1 Definition of expectation 6 2 Convergence  69 
An Easy Strong Law  71 
rule for measures Probability theory supplements that with the multipli  83 
Martingales bounded in C2  110 
Uniform Integrability  126 
5 Martingale proof of the Strong Law 14 6 Doobs Sub  150 
CHARACTERISTIC FUNCTIONS  172 
The Central Limit Theorem  185 
Appendix to Chapter 3  205 
Appendix to Chapter 0  214 
243  
Common terms and phrases
7rsystem aalgebra Aset absolutely continuous absolutely continuous relative algebra Borel BorelCantelli Lemma bounded in Cl Chapter choose conditional expectation Convergence Theorem converges a.s. course dsystem deduce define definition denote disjoint distribution function Doob's E(Xn elements equivalent example Exercise exists a.s. Fatou Lemma Fatou's Lemma finite Fubini's Theorem given Hence IID RVs independent random variables independent RVs indicator function infinitely integral intuitive Jensen's inequality Kolmogorov's Kronecker's Lemma Lebesgue measure Let Xn liminf limsup lira sup Markov chain martingale martingale relative measure space monotonicity nonnegative notation Note null obtain obvious previsible process Prob(R probability measure probability triple prove RadonNikodym theorem real numbers Section sequence Fn sequence of independent shows stopping Strong Law subaalgebra submartingale subsets supermartingale Suppose that Xn surely THEOREM Let Tower Property UI martingale unique Xni Xn(u