A First Look at Rigorous Probability Theory
World Scientific, 2000 - 177 páginas
This textbook is an introduction to rigorous probability theory using measure theory. It provides rigorous, complete proofs of all the essential introductory mathematical results of probability theory and measure theory. More advanced or specialized areas are entirely omitted or only hinted at. For example, the text includes a complete proof of the classical central limit theorem, including the necessary continuity theorem for characteristic functions, but the more general Lindeberg central limit theorem is only outlined and is not proved. Similarly, all necessary facts from measure theory are proved before they are used, but more abstract or advanced measure theory results are not included. Furthermore, measure theory is discussed as much as possible purely in terms of probability, as opposed to being treated as a separate subject which must be mastered before probability theory can be understood.
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The need for measure theory
Further probabilistic foundations
Inequalities and laws of large numbers
Distributions of random variables
Stochastic processes and gambling games
Discrete Markov chains
Some further probability results
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a-algebra absolutely continuous Additional exercises algebra aperiodic Borel probability measure Borel sets Brownian motion candidate density Central Limit Theorem characteristic function claimed collection compute consider continuity of probabilities continuity theorem continuous-time converges weakly Corollary countably additive cr-algebra define definition diffusion discrete disjoint E(Xn E(XY equal equation equivalence expected value follows Furthermore Fx(x given Hence Hint independent random variables inequality infinite integral Intuitively large numbers law of large Lebesgue measure Lemma Let Xn lim inf linear Markov chain martingale mathematical measure on 0,1 measure theory monotone convergence theorem Mx(s P(Xn positive recurrent probability measure probability theory probability triple properties Proposition Prove real numbers result Section summary sequence Similarly simple random variable simple symmetric random stationary distribution stochastic process submartingale subsets supn Suppose symmetric random walk transition probabilities union unique Var(X variance weak convergence Xn(uj