Probability EssentialsSpringer Science & Business Media, 6 dic 2012 - 250 pagine We present here a one-semester course on Probability Theory. We also treat measure theory and Lebesgue integration, concentrating on those aspects which are especially germane to the study of Probability Theory. The book is intended to fill a current need: there are mathematically sophisticated stu dents and researchers (especially in Engineering, Economics, and Statistics) who need a proper grounding in Probability in order to pursue their primary interests. Many Probability texts available today are celebrations of Prob ability Theory, containing treatments of fascinating topics to be sure, but nevertheless they make it difficult to construct a lean one semester course that covers (what we believe are) the essential topics. Chapters 1-23 provide such a course. We have indulged ourselves a bit by including Chapters 24-28 which are highly optional, but which may prove useful to Economists and Electrical Engineers. This book had its origins in a course the second author gave in Perugia, Italy, in 1997; he used the samizdat "notes" of the first author, long used for courses at the University of Paris VI, augmenting them as needed. The result has been further tested at courses given at Purdue University. We thank the indulgence and patience of the students both in Perugia and in West Lafayette. We also thank our editor Catriona Byrne, as weil as Nick Bingham for many superb suggestions, an anonymaus referee for the same, and Judy Mitchell for her extraordinary typing skills. Jean Jacod, Paris Philip Protter, West Lafayette Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . |
Sommario
| 2 | |
Conditional Probability and Independence | 11 |
Probabilities on a Countable Space | 17 |
Construction of a Probability Measure | 40 |
Integration with Respect to a Probability Measure | 47 |
Independent Random Variables | 61 |
Probability Distributions on | 83 |
Properties of Characteristic Functions | 107 |
Gaussian Random Variables The Normal and the Multi | 120 |
Convergence of Random Variables | 137 |
Weak Convergence and Characteristic Functions | 163 |
L2 and Hilbert Spaces | 185 |
Martingales 207 | 206 |
Martingale Convergence Theorems 225 | 224 |
The RadonNikodym Theorem | 239 |
Characteristic Functions | 110 |
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assume Binomial Borel measurable Borel sets bounded Cauchy Central Limit Theorem characteristic function conditional expectation continuous functions converges in distribution converges weakly Corollary countable covariance matrix deduce defined Definition denote density f distribution function Dominated Convergence Theorem E{X² E{Xn E{XY equivalent example Exercises for Chapter exists exponential finite Fn(x function F Gamma given hence Hilbert space Hint implies independent inequality infn integrable Large Numbers Law of Large Lebesgue measure Let f let G Let Sn lim inf lim sup limn linear martingale Monotone Convergence Theorem nonnegative Note open sets P(An P(Xn pairwise disjoint Poisson positive probability measure probability space Proof R-valued result sequence of random Show Strong Law submartingale subset Suppose unique valued random variables variance vector Xn converges Y₁ σ²