PTE - 5th Edition Probability: Theory and Examples. 5th Edition1. Measure Theory1. Probability Spaces2. Random Variables4. Properties of the Integral6.
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Expected Value7. Product Measures, Fubini's Theorem2. Laws of Large Numbers1. Weak Laws of Large Numbers3. Borel-Cantelli Lemmas4.
Strong Law of Large Numbers5. Convergence of Random Series.6. Renewal Theory.7. Large Deviations.3.
Central Limit Theorems1. The De Moivre-Laplace Theorem2. Weak Convergence3. Characteristic Functions4. Central Limit Theorems5. Local Limit Theorems.6. Poisson Convergence7.
Poisson Processes8. Stable Laws.9. Infinitely Divisible Distributions.10. Limit Theorems in R d.4. Conditional Expectation2.
Martingales, Almost Sure Convergence3. Doob's Inequality, L p Convergence5. Square Integrable Martingales (was Subsection 5.4.1)6. Uniform Integrability, Convergence in L 17. Backwards Martingales8.
Optional Stopping Theorems9. Combinatorics of Simple Random Walk5. Markov Chains1. Construction, Markov Properties3. Recurrence and Transience4.
Recurrence of Random Walks5. Stationary Measures6. Asymptotic Behavior7. Periodicity, Tail σ-field.8.
General State Space.6. Ergodic Theorems1.
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Definitions and Examples2. Birkhoff's Ergodic Theorem3. A Subadditive Ergodic Theorem5. Brownian Motion1. Definition and Construction2. Markov Property, Blumenthal's 0-1 Law3.
Stopping Times, Strong Markov Property4. Maxima and Zeros5. Ito's formula.8. Brownian Embeddings and Applications1. Donsker's Theorem2. CLTs for Martingales3.
CLTs for Stationary Sequences4. Empirical Distributions, Brownian Bridge5. Laws of the Iterated Logarithm9. Multidimensional Brownian Motion1. Heat Equation3.
Inhomogenous Heat Equation4. Feynman-Kac Fromula5. Dirichlet Problem6. Green's Functions and Potential Kernels7. Poisson's Equation8.
Schrodinger EquationAppendix: Measure Theory1. Caratheodary's Extension Theorem2. Which sets are measurable?3. Kolmogorov's Extension Theorem4. Radon-Nikodym Theorem5.
Differentiating Under the Integral.
Probability theory arises in the modelling of a variety of systems where the understanding of the 'unknown' plays a key role, such as population genetics in biology, market evolution in financial mathematics, and learning features in game theory. It is also very useful in various areas of mathematics, including number theory and partial differential equations. The course introduces the basic mathematical framework underlying its rigorous analysis, and is therefore meant to provide some of the tools which will be used in more advanced courses in mathematics.The first part of the course provides a review of measure theory from Integration Part A, and develops a deeper framework for its study. Then we proceed to develop notions of conditional expectation, martingales, and to show limit results for the behaviour of these martingales which apply in a variety of contexts.
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Brzezniak and T. Zastawniak, Basic stochastic processes. A course through exercises.
Springer Undergraduate Mathematics Series. (Springer-Verlag London, Ltd., 1999) more elementary than D. Williams' book, but can provide with a complementary first reading. M. Capinski and E.
Kopp, Measure, integral and probability, Springer Undergraduate Mathematics Series. (Springer-Verlag London, Ltd., second edition, 2004). R. Durrett, Probability: Theory and Examples (Second Edition Duxbury Press, Wadsworth Publishing Company, 1996). A. Etheridge, A Course in Financial Calculus, (Cambridge University Press, 2002).
J. Neveu, Discrete-parameter Martingales (North-Holland, Amsterdam, 1975).
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