C8.6 Limit Theorems and Large Deviations in Probability (2021-22)
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- Lecturer: Profile: Zhongmin Qian
Course information
General Prerequisites:
Part A Probability and Part A Integration are required. B8.1 (Measure, Probability and Martingales), B8.2 (Continuous Martingales and Stochastic Calculus) and C8.1 (Stochastic Differential Equations) are desirable, but not essential.
Course Term: Hilary
Course Lecture Information: 16 lectures.
Course Weight: 1
Course Level: M
Assessment Type: Written Examination
Course Overview:
The convergence theory of probability distributions on path space is an essential part of modern probability and stochastic analysis allowing the development of diffusion approximations and the study of scaling limits in many settings. The theory of large deviation is an important aspect of limit theory in probability as it enables a description of the probabilities of rare events. The emphasis of the course will be on the development of the necessary tools for proving various limit results and the analysis of large deviations which have universal value. These topics are fundamental within probability and stochastic analysis and have extensive applications in current research in the study of random systems, statistical mechanics, functional analysis, PDEs, quantum mechanics, quantitative finance and other applications.
Learning Outcomes:
The students will understand the notions of convergence of probability laws, and the tools for proving associated limit theorems. They will have developed the basic techniques for the establishing large deviation principles and be able to analyze some fundamental examples.
Course Synopsis:
1) (2 lectures) We will recall metric spaces, and introduce Polish spaces, and probability measures on metric spaces. Weak convergence of probability measures and tightness, Prohorov's theorem on tightness of probability measures, Skorohod's representation theorem for weak convergence.
2) (2 lectures) The criterion of pre-compactness for distributions on continuous path spaces, martingales and compactness.
3) (4 hours) Skorohod's topology and metric on the space \(D[0,\infty)\) of right-continuous paths with left limits, basic properties such as completeness and separability, weak convergence and pre-compacness of distributions on \(D[0,\infty)\). D. Aldous' pre-compactness criterion via stopping times.
4) (4 lectures) First examples - Cramér's theorem for finite dimensional distributions, Sanov's theorem. Schilder's theorem for the large deviation principle for Brownian motion in small time, law of the iterated logarithm for Brownian motion.
5) (4 lectures) General tools in large deviations. Rate functions, good rate functions, large deviation principles, weak large deviation principles and exponential tightness. Varadhan's contraction principle, functional limit theorems.
2) (2 lectures) The criterion of pre-compactness for distributions on continuous path spaces, martingales and compactness.
3) (4 hours) Skorohod's topology and metric on the space \(D[0,\infty)\) of right-continuous paths with left limits, basic properties such as completeness and separability, weak convergence and pre-compacness of distributions on \(D[0,\infty)\). D. Aldous' pre-compactness criterion via stopping times.
4) (4 lectures) First examples - Cramér's theorem for finite dimensional distributions, Sanov's theorem. Schilder's theorem for the large deviation principle for Brownian motion in small time, law of the iterated logarithm for Brownian motion.
5) (4 lectures) General tools in large deviations. Rate functions, good rate functions, large deviation principles, weak large deviation principles and exponential tightness. Varadhan's contraction principle, functional limit theorems.
Section outline
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Will take place on Thursday, 19 May, 9- 10:30, in Room L5
- Review some material on weak convergence of probabilities, and tightness of a family of continuous processes.
- Look at year 2019 or/and 2020 paper.
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3rd Consultation Session
Will takes place on 26 May, Thursday, from 10 to 11 in L5.
Please send me your questions you want to discuss in advance (a day before) via email.________________________________________________________________________________Microsoft Teams meetingJoin on your computer or mobile appClick here to join the meeting________________________________________________________________________________Microsoft Teams meetingJoin on your computer or mobile appClick here to join the meeting -
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