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Mathematical Foundations of Probability and Statistics

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Poster

A/Prof Ben Goldys
The University of New South Wales

Synopsis

Probabilistic arguments have been known for thousands of years - some of them can be found in the Old Testament and Talmud. In 1933 A. N. Kolmogorov demonstrated that measure theory and analysis provide the proper language to study probability in a rigorous way. Nowadays, the mathematical theory of probability is applied in the areas as diverse as statistics, mathematical finance, genetics and quantum field theory. It is also an important tool for mathematicians working in differential equations, differential geometry and graph theory. In this course we will introduce the basic concepts of the mathematical theory of probability and will discuss some of its applications.

The course will cover the following topics:

  1. Probability measures, methods to introduce a probability measure
  2. Random variables, distribution functions, independence
  3. Expectation as the abstract Lebesgue integral
  4. Some important inequalities and uniform integrability
  5. Convergence of random variables, the Borel-Cantelli Lemma
  6. The Strong Law of Large Numbers
  7. The Radon-Nikodym theorem and densities of probability measures
  8. Conditional expectations and regular conditional probabilities
  9. L2-space of random variables and Gaussian systems
  10. Weak convergence of measures and characteristic functions
  11. The Central Limit Theorem
  12. Martingales

Contact hours

28 hours of hours spread over four weeks, plus consultation as required.

Prerequisites

Essential: a course in Calculus, understanding the concepts of limits of sequences and functions

Helpful: a basic course in Probability.

Resources

Lecture notes will be distributed but part of them will be very brief. Additional reading is strongly recommended. The titles below are listed in order of increasing difficulty (approximately):

  1. Capiński M. and Kopp E.: Measure, Integral and Probability (Second edition), Springer 2005
  2. Capiński M. and Zastawniak T.: Probability Through Problems, Springer
  3. Shiryaev A.N.: Probability, Springer
  4. Durrett R.: Probability: Theory and Examples (second edition) Duxbury Press
  5. Billingsley P.: Probability and Measure, Wiley
  6. Dudley R. M.: Real Analysis and Probability, Cambridge UP
The book by Shiryaev is strongly recommended.

About Ben Goldys

I am an Associate Professor at the School of Mathematics and Statistics at UNSW. My research is focused on mathematical analysis of systems evolving randomly in space and time. Such systems arise in Fluid Dynamics, Climate Science, Physics, Finance and Population Genetics. The main tool for their analysis is the theory of partial differential equations perturbed by random force and analysis on infinite-dimensional spaces. My recent collaborative projects are in the area of stochastic partial differential equations on manifolds and are motivated by the physics of magnetic materials.