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Department of Mathematical Sciences

# MATH3211/4131 Probability III/IV

Randomness is an essential feature of many models of real-world phenomena. Examples include locations of failures in a power grid, traffic fluctuations in the internet, flaws in ultra-pure materials, statistical behaviour of polymers to list just a few. While classical probability theory for sums of independent (or weakly dependent) variables can be used to describe some of these models, analysis and prediction of more dependent systems, especially for extreme events (like natural disasters, stock market crashes and bubbles, collective behaviour of social networks) require different approaches.

The purpose of this course is to attempt to bring your everyday intuition and common sense into agreement with the laws of probability and to explore how they can be used to analyse various applied models. We shall use elementary methods wherever possible, while discussing a variety of classical and modern applications of the subject ranging from random walks and renewal theory to records, extreme values and non-Gaussian limit results.

## Outline of Course

Aim: This module continues on from Probability II. It is designed to build a logical structure on probabilistic intuition; to study classical results such as the Strong Law of Large Numbers, the Central Limit Theorem, and some applications; to discuss some modern developments in the subject. Students completing this course should be equipped to read for themselves much of the vast literature on applications to biology, physics, chemistry, mathematical finance, engineering and many other fields.

### Term 1

• Introductory examples: from finite to infinite spaces.
• Coin tossing and trajectories of random walks: reflection principle, ballot theorem; returns to the origin; excursions; arcsine laws.
• Discrete renewal theory and its applications.
• Limit theorems and convergence: modes of convergence for random variables; weak law of large numbers; central limit theorem; Borel-Cantelli lemma and its applications; Kolmogorov's strong law of large numbers.

### Term 2

• Order statistics: order variables and their distributions; exchangeability; limiting behaviour of extrema; some applications.
• Non-classical limits and their applications: convergence to Poisson and exponential distributions; applications, extreme value theory.
• Stochastic order and its applications: coupling bounds on total variation distance.
• Topics chosen from: zero-one laws; percolation; random graphs; large deviations; Gibbs distributions and phase transitions; randomized algorithms; probabilistic combinatorics.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook: MATH3211, MATH4131.