A First Look at Rigorous Probability Theory

Höfundur: Jeffrey S Rosenthal (Útgáfa: 2)
A First Look at Rigorous Probability Theory

Kaup valmöguleikar

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.

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Útgefandi
World Scientific Publishing
ISBN
9789813105706
Print ISBN
Format
Page Fidelity (PDF)
Útgáfa
2
Höfundar
Jeffrey S Rosenthal
Tungumál
English
Útgefið
20000420
Prent takmörkun á líftíma
100
Prent takmörkun
2
Afritunar takmörkun
2

Kaflar

  • Contents
  • Preface.
  • 1. The need for measure theory. This introductory section is directed primarily to those
  • 1.1. Various kinds of random variables.
  • 1.2. The uniform distribution and non-measurable sets.
  • 1.3. Additional exercises.
  • 1.4. Section summary.
  • 2. Probability triples.
  • 2.1. Basic definition.
  • 2.2. Discrete probability spaces.
  • 2.3. Constructing Lebesgue measure.
  • 2.4. The extension theorem.
  • 2.5. More on Lebesgue measure.
  • 2.6. Coin tossing and other measures.
  • 2.7. Additional exercises.
  • 2.8. Section summary.
  • 3. Further probabilistic foundations.
  • 3.1. Random variables.
  • 3.2. Independence.
  • 3.3. Continuity of probabilities.
  • 3.4. Limit events.
  • 3.5. Tail fields.
  • 3.6. Additional exercises.
  • 3.7. Section summary.
  • 4. Expected values.
  • 4.1. Simple random variables.
  • 4.2. General non-negative random variables.
  • 4.3. Arbitrary random variables.
  • 4.4. The integration connection.
  • 4.5. Additional exercises.
  • 4.6. Section summary,
  • 5. Inequalities and laws of large numbers.
  • 5.1. Weak law of large numbers.
  • 5.2. Strong law of large numbers.
  • 5.3. Eliminating the moment conditions.
  • 5.4. Additional exercises.
  • 5.5. Section summary.
  • 6. Distributions of random variables.
  • 6.1. Change of variable theorem.
  • 6.2. Examples of distributions.
  • 6.3. Additional exercises.
  • 6.4. Section summary.
  • 7. Stochastic processes and gambling games.
  • 7.1. A first existence theorem.
  • 7.2. Gambling and gambler's ruin.
  • 7.3. Gambling policies.
  • 7.4. Additional exercises.
  • 7.5. Section summary.
  • 8. Discrete Markov chains.
  • 8.1. A Markov chain existence theorem.
  • 8.2. Transience, recurrence, and irreducibility.
  • 8.3. Stationary distributions and convergence.
  • 8.4. Existence of stationary distributions.
  • 8.5. Additional exercises.
  • 8.6. Section summary.
  • 9. Some further probability results.
  • 9.1. Limit theorems.
  • 9.2. Differentiation and expectation.
  • 9.3. Moment generating functions and large deviations.
  • 9.4. Convolution and Fubini's Theorem.
  • 9.5. Additional exercises.
  • 9.6. Section summary.
  • 10. Weak convergence.
  • 10.1. Additional exercises.
  • 10.2. Section summary.
  • 11. Characteristic functions.
  • 11.1. The continuity theorem.
  • 11.2. The Central Limit Theorem.
  • 11.3. Generalisations of the Central Limit Theorem.
  • 11.4. Method of moments.
  • 11.5. Additional exercises.
  • 11.6. Section summary.
  • 12. Decomposition of probability laws.
  • 12.1. Additional exercises.
  • 12.2. Section summary.
  • 13. Conditional probability and expectation.
  • 13.1. Additional exercises.
  • 13.2. Section summary.
  • 14. Martingales.
  • 14.1. Stopping times.
  • 14.2. Convergence.
  • 14.3. Other results.
  • 14.4. Additional exercises.
  • 14.5. Section summary.
  • 15. Introduction to other stochastic processes.
  • 15.1. Kolmogorov Existence Theorem.
  • 15.2. Markov chains on general state spaces.
  • 15.3. Continuous-time processes.
  • 15.4. Brownian motion as a limit.
  • 15.5. Existence of Brownian motion.
  • 15.6. Diffusions and stochastic integrals.
  • 15.7. Ito's Lemma.
  • 15.8. The Black-Scholes equation.
  • 15.9. Section summary.
  • Appendix: Mathematical Background.
  • A.1. Sets and functions.
  • A.2. Countable sets.
  • A.3. Epsilons and Limits.
  • A.4. Infimums and supremums.
  • A.5. Equivalence relations.
  • Bibliography.
  • B.1. Background in real analysis.
  • B.2. Undergraduate-level probability.
  • B.3. Graduate-level probability.
  • B.4. Pure measure theory.
  • B.5. Stochastic processes.
  • B.6. Mathematical finance.
  • Index