This introduction can be used, at the beginning graduate level, for a one-semester course on probability theory or for self-direction without benefit of a formal course; the measure theory needed is developed in the text. It will also be useful for students and teachers in related areas such as finance theory, electrical engineering, and operations research. The text covers the essentials in a directed and lean way with 28 short chapters, and assumes only an undergraduate background in mathematics. Readers are taken right up to a knowledge of the basics of Martingale Theory, and the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference.
Inhaltsverzeichnis
1 Introduction. - 2 Axioms of Probability. - 3 Conditional Probability and Independence. - 4 Probabilities on a Finite or Countable Space. - 5 Random Variables on a Countable Space. - 6 Construction of a Probability Measure. - 7 Construction of a Probability Measure on R. - 8 Random Variables. - 9 Integration with Respect to a Probability Measure. - 10 Independent Random Variables. - 11 Probability Distributions on R. - 12 Probability Distributions on Rn. - 13 Characteristic Functions. - 14 Properties of Characteristic Functions. - 15 Sums of Independent Random Variables. - 16 Gaussian Random Variables (The Normal and the Multivariate Normal Distributions). - 17 Convergence of Random Variables. - 18 Weak Convergence. - 19 Weak Convergence and Characteristic Functions. - 20 The Laws of Large Numbers. - 21 The Central Limit Theorem. - 22 L2 and Hilbert Spaces. - 23 Conditional Expectation. - 24 Martingales. - 25 Supermartingales and Submartingales. - 26 Martingale Inequalities. - 27 Martingale Convergence Theorems. - 28 The Radon-Nikodym Theorem. - References.
