A First Look At Rigorous Probability TheoryWorld Scientific Publishing Company, 20 de abr. de 2000 - 192 páginas This textbook is an introduction to probability theory using measure theory. It is designed for graduate students in a variety of fields (mathematics, statistics, economics, management, finance, computer science, and engineering) who require a working knowledge of probability theory that is mathematically precise, but without excessive technicalities. The text provides complete proofs of all the essential introductory results. Nevertheless, the treatment is focused and accessible, with the measure theory and mathematical details presented in terms of intuitive probabilistic concepts, rather than as separate, imposing subjects. The text strikes an appropriate balance, rigorously developing probability theory while avoiding unnecessary detail. |
Conteúdo
1 The need for measure theory This introductory section is directed primarily to those | 1 |
2 Probability triples | 6 |
3 Further probabilistic foundations | 21 |
4 Expected values | 32 |
5 Inequalities and laws of large numbers | 43 |
6 Distributions of random variables | 52 |
7 Stochastic processes and gambling games | 58 |
8 Discrete Markov chains | 68 |
11 Characteristic functions | 102 |
12 Decomposition of probability laws | 118 |
13 Conditional probability and expectation | 124 |
14 Martingales | 131 |
15 Introduction to other stochastic processes | 140 |
Mathematical Background | 160 |
Bibliography | 168 |
| 171 | |
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Termos e frases comuns
absolutely continuous Additional exercises algebra aperiodic Borel sets Borel-measurable function Brownian motion candidate density Central Limit Theorem characteristic functions collection compute consider continuity of probabilities converges weakly Corollary countably additive define definition discrete disjoint E(Xn E(Xo equation equivalence exists expected value follows function f Furthermore given Hence Hint inequality infinite integral interval Intuitively large numbers law of large Lebesgue measure Lemma let Q Let Xn lim sup limn linear Markov chain martingale measure on 0,1 measure theory monotone convergence theorem non-negative o-algebra P(An P(lim P(Xn positive recurrent probability measure probability theory probability triple Proposition Prove real numbers result Section summary sequence Similarly simple random variable simple symmetric random stationary distribution stochastic process submartingale subsets Suppose symmetric random walk transition probabilities triple Q Var(X variance weak convergence