Dr Beatrice Acciaio and Dr Luciano Campi

This course is available on the MPhil/PhD in Statistics. This course is available with permission as an outside option to students on other programmes where regulations permit.

The availability as an outside option requires a demonstration of sufficient background in mathematics and statistics and is at the discretion of the instructor.

This course provides theoretical and axiomatic foundations of probability and mathematical statistics,

and is intended for PhD students in the Statistics department. In particular, the following topics will be covered:

1. Measure spaces; Caratheodory extension theorem; Borel-Cantelli lemmas.

2. Random variables; monotone-class theorem; different kinds of convergence.

3. Kolmogorov’s 0-1 law; construction of Lebesgue integral.

4. Monotone convergence theorem; Fatou's lemmas; dominated convergence theorem.

5. Expectation; L^p spaces; uniform integrability.

6. Characteristic functions; Levy inversion formula; Levy convergence theorem; CLT.

7. Principle and basis for statistical inference: populations and samples, decision theory, basic

measures for estimators.

8. Estimation: U and V statistics, unbiased estimators, MVUE, MLE.

9. Hypothesis testing: Neyman-Pearson lemma, UMP, confidence sets.

10. Product measures; conditional expectation.

20 hours of lectures and 10 hours of seminars in the MT.

Students will be expected to produce 9 problem sets in the MT.

Weekly problem sets that are discussed in subsequent seminars. The coursework that will be used for summative assessment will be chosen from a subset of these problems.

1. Williams, D. (1991). Probability with Martingales. Cambridge University Press.
2. Durrett, R. (2019). Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics.
3. Shao, J. (2007). Mathematical Statistics. Springer Texts in Statistics.
4. Keener, R. (2010). Theoretical Statistics. Springer Texts in Statistics.

Exam (70%, duration: 3 hours, reading time: 10 minutes) in the January exam period.
Coursework (30%).

Three of the homework problem sets will be submitted and marked as assessed coursework.