Structure

This exam is a four-four exam on probability theory. It is assumed that all candidates will have a background corresponding to Statistics 521, 522, 523. The exam will typically consist of 4-8 questions on the following topics:

• Measure theory
• Integration
• Maximal inequalities
• Strong and weak limit theorems for independent random variables
• Convergence in distribution and weak convergence of laws
• Conditional expectation
• Martingales, martingale convergence theorems, and optional sampling
• Brownian motion

A study guide for each of these topics and references are given below.

Time

This exam is given once a year. In the past it has been given during the summer months, usually in June or August.

Study Guide and References

MEASURE THEORY: Mappings and sigma-fields; measurable functions; modes of convergence; induced measures; decomposition of signed measures; Radon Nikodym theorem; measures on product spaces; Fubini's theorem; Lebesgue's decomposition theorem.

• Chung, K.L. A Course in Probability Theory, Chapters 1-3.
• Billingsley, P. Probability and Measure, Wiley, New York, Chapter 2 and Appendix.
• Shorack, G.R. Probability for Statisticians. Chapters 1-5.

INTEGRATION: The Lebesgue integral; monotone and dominated convergence theorems, Fatou's lemma; absolute continuity of the integral; inequalities; modes of convergence; uniform integrability; Vitali's theorem.

• Chung, K.L. A Course in Probability Theory, Chapters 3-4.
• Billingsley, P. Probability and Measure, Wiley, New York, Chapters 3-4.
• Shorack, G.R. Probability for Statisticians. Chapter 3.

MAXIMAL INEQUALITIES: Lévy's inequalities; Ottaviani-Skorohod inequality; Kolmogorov's inequality; Hoffmann-Jorgensen inequalities.

• Van der Vaart and Wellner, Weak Convergence and Empirical Processes, Springer-Verlag, New York, Appendix A.1
• Shorack, G.R. Probability for Statisticians. Chapter 10.

STRONG AND WEAK LIMIT THEOREMS FOR INDEPENDENT RANDOM VARIABLES: Strong and weak laws of large numbers; behavior of the maximal term; convergence in L₁ and L_p, p>1; convergence of series; three-series theorem.

• Chung, K.L. A Course in Probability Theory, Chapter 5.
• Billingsley, P. Probability and Measure, Wiley, New York, Chapters 4.
• Shorack, G.R. Probability for Statisticians. Chapter 10.

CONVERGENCE IN DISTRIBUTION: Tightness and Helly-Bray; Helly selection theorem; Mann-Wald theorem; Lindeberg-Lévy and Lindeberg-Feller CLT's; Berry-Esseen theorem; characteristic functions; infinite divisible laws.

• Chung, K.L. A Course in Probability Theory, Chapters 6-7.
• Billingsley, P. Probability and Measure, Wiley, New York, Chapter 5.
• Shorack, G.R. Probability for Statisticians. Chapters 11, 13 - 15.

CONDITIONAL EXPECTATION: Definition of conditional expectation and conditional probabilities; properties of conditional expectation; Regular conditional expectation; Conditional Expectation as least squares predictor in L₂.

• Chung, K.L. A Course in Probability Theory, Chapter 9.
• Billingsley, P. Probability and Measure, Wiley, New York, Chapter 6.
• Shorack, G.R. Probability for Statisticians. Chapter 8.

MARTINGALES: Definitions and basic properties; simple optional sampling; upcrossing inequalities; martingale and sub-martingale convergence theorems; decompositions of a sub-martingale; optional sampling.

• Chung, K.L. A Course in Probability Theory, Chapter 9.
• Billingsley, P. Probability and Measure, Wiley, New York, Chapter 6.
• Shorack, G.R. Probability for Statisticians. Chapter 18.

BROWNIAN MOTION: Existence as a process with values in C[0,1]; transformations; strong Markov property; barrier crossings; transformations and basic sample path properties.

• Billingsley, P. Probability and Measure, Wiley, New York, Chapter 7.
• Dudley, R. M. Real Analysis and Probability, Chapman and Hall, NY, Chapter 12.
• Shorack, G.R. Probability for Statisticians. Chapter 12.