Course Materials
Main content blocks
- Lecturer: Profile: Matthias Winkel
Moment generating functions and applications. Statements of the continuity and uniqueness theorems for moment generating functions. Characteristic functions (definition only). Convergence in distribution and convergence in probability. Weak law of large numbers and central limit theorem for independent identically distributed random variables. Strong law of large numbers (proof not examinable).
Discrete-time Markov chains: definition, transition matrix, n-step transition probabilities, communicating classes, absorption, irreducibility, periodicity, calculation of hitting probabilities and mean hitting times. Recurrence and transience. Invariant distributions, mean return time, positive recurrence, convergence to equilibrium (proof not examinable), ergodic theorem (proof not examinable). Random walks (including symmetric and asymmetric random walks on \(Z\), and symmetric random walks on \(Z^d\).
Poisson processes in one dimension: exponential spacingβs, Poisson counts, thinning and superposition.
Section outline
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Lectures are based on the Lecture Notes π provided below. The lectures are whiteboard-based and are I encourage taking notes. There are only two lectures, where I also use slides, First Slides πΌοΈ in the review part of the first lecture on Chapter 1 and Last Slides πΌοΈ in the final part of the last lecture on Section 7.5.
There are four problem sheets. π Sheet 1 covers Chapters 1-2, π Sheet 2 covers Chapters 3-4, π Sheet 3 covers Chapter 5 and π Sheet 4 covers Chapters 6-7.
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These appendices contain a discussion of the theoretical background to the Prelims Probability course, establishing and exploiting connections to the Analysis courses in Prelims and Part A. This additional material is non-examinable in Part A Probability.
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This appendix includes a discussion of some more theoretical background to the Part A Probability course. This additional material is non-examinable in Part A Probability.