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Modern and measure-theory based, this text is intended primarily for the first-year graduate course in probability theory. The book focuses attention on examples while developing theory. There is an emphasis on results that can be used to solve problems in the hopes that those who apply probability to work will find this a useful reference.
Benefits:
INTRODUCTORY LECTURE.
1. LAWS OF LARGE NUMBERS.
Basic Definitions. Random Variables. Expected Value. Independence. Weak Laws of Large Numbers. Borel-Cantelli Lemmas. Strong Law of Large Numbers. Convergence of Random Series. Large Deviations.
2. CENTRAL LIMIT THEOREMS.
The De Moivre-Laplace Theorem. Weak Convergence. Characteristic Functions. Central Limit Theorems. Local Limit Theorems. Poisson Convergence. Stable Laws. Infinitely Divisible Distributions. Limit theorems in Rd.
3. RANDOM WALKS.
Stopping Times. Recurrence. Visits to 0, Arcsine Laws. Renewal Theory.
4. MARTINGALES.
Conditional Expectation. Martingales, Almost Sure Convergence. Examples. Doob's Inequality, Lp Convergence. Uniform Integrability, Convergence in L1.Backwards Martingales. Optional Stopping Theorems. Optional Stopping Theorems.
5. MARKOV CHAINS.
Definitions and Examples. Extensions of the Markov Property. Recurrence and Transience. Stationary Measures. Asymptotic Behavior. General State Space.
6. ERGODIC THEOREMS.
Definitions and Examples. Birkhoff's Egodic Theorem. Recurrence. Mixing. Entropy. A Subadditive Ergodic Theorem. Applications.
7. BROWNIAN MOTION.
Definition and Construction. Markov Property, Blumenthal's 0-1 Law. Stopping Times, Strong Markov Property. Maxima and Zeros. Martingales. Donsker's Theorem. CLT's for Dependent Variables. Empirical Distributions, Brownian Bridge. Laws of the Iterated Logarithm.
APPENDIX: MEASURE THEORY.
REFERENCES.
NOTATION.
NORMAL TABLE.
INDEX.
Modern and measure-theory based, this text is intended primarily for the first-year graduate course in probability theory. The book focuses attention on examples while developing theory. There is an emphasis on results that can be used to solve problems in the hopes that those who apply probability to work will find this a useful reference.
Benefits:
Table of Contents
INTRODUCTORY LECTURE.
1. LAWS OF LARGE NUMBERS.
Basic Definitions. Random Variables. Expected Value. Independence. Weak Laws of Large Numbers. Borel-Cantelli Lemmas. Strong Law of Large Numbers. Convergence of Random Series. Large Deviations.
2. CENTRAL LIMIT THEOREMS.
The De Moivre-Laplace Theorem. Weak Convergence. Characteristic Functions. Central Limit Theorems. Local Limit Theorems. Poisson Convergence. Stable Laws. Infinitely Divisible Distributions. Limit theorems in Rd.
3. RANDOM WALKS.
Stopping Times. Recurrence. Visits to 0, Arcsine Laws. Renewal Theory.
4. MARTINGALES.
Conditional Expectation. Martingales, Almost Sure Convergence. Examples. Doob's Inequality, Lp Convergence. Uniform Integrability, Convergence in L1.Backwards Martingales. Optional Stopping Theorems. Optional Stopping Theorems.
5. MARKOV CHAINS.
Definitions and Examples. Extensions of the Markov Property. Recurrence and Transience. Stationary Measures. Asymptotic Behavior. General State Space.
6. ERGODIC THEOREMS.
Definitions and Examples. Birkhoff's Egodic Theorem. Recurrence. Mixing. Entropy. A Subadditive Ergodic Theorem. Applications.
7. BROWNIAN MOTION.
Definition and Construction. Markov Property, Blumenthal's 0-1 Law. Stopping Times, Strong Markov Property. Maxima and Zeros. Martingales. Donsker's Theorem. CLT's for Dependent Variables. Empirical Distributions, Brownian Bridge. Laws of the Iterated Logarithm.
APPENDIX: MEASURE THEORY.
REFERENCES.
NOTATION.
NORMAL TABLE.
INDEX.