Learning objectives
Since lacking a more advanced course in probability, this course gives a quick but quite comprehensive collection of the main themes of modern probability theory, with a particular attention towards martingale theory and Markov chains.
Prerequisites
<ol>
<li>Elementi di probabilità</li>
<li>Teoria della misura e dell'integrazione</li>
</ol>
Course unit content
Probability spaces. Sigma-fields. Random variables. Induced law. Absolute continuity. Properties of cumulative function. Measurability lemma. Skorokhod theorem. Mathematical expectation. Monotone convergence. Fatou lemmas, dominated and Scheffé convergence. Jensen inequality. L^p norm. Borel-Cantelli lemmas. Inclusion-exclusion principle.<br />
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Markov inequality, Chebyshev inequality, with exponential-optimized version. Legendre transform. Type of convergence for sequences of random variables. Uniform integrability.<br />
<br />
Conditional expectation. Existence in L^2 (by projection) and in L^1 (by density). Properties.<br />
<br />
Independence of sigma-fileds. Strong law of large numbers (through Garsia lemma, uniform integrability, Kolmogorov's 0-1 law).<br />
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Weak convergence. Characteristic function. Lévy theorem. Thightness. Central limit theorem.<br />
<br />
Martingales. Discrete stochastic integral. Stopping times. Stopped processes. Optional stopping theorem. Martingale convergence theorem. Closed and closable martingales. Lévy downward theorem and Law of Large Numbers (again).<br />
<br />
Simple simmetric random walk. First and last time in 0. Reflection principle.<br />
<br />
Radon-Nikodym theorem. Hewitt-Savage theorem.<br />
<br />
L^2 martingales, Doob decomposition, quadratic variation. Convergence theorem for L^2 martingales.<br />
<br />
Branching processes. Extintion probability. Exponential scaling.<br />
<br />
Polya urn and urn processes. Some a.s. convergence results.<br />
<br />
Markov chains. Strong and weak Markov property. State classification. Invariant measures.<br />
<br />
Large deviations (shortly).
Full programme
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Bibliography
<ol>
<li>D. Williams - Probability with Martingales - Cambridge University Press - 1991</li>
<li>Z. Brzezniak, T. Zastawniak - Basic Stochastic Processes - Springer 1999</li>
<li>J. Jacod, P. Protter - Probability Essentials - Springer 1999</li>
</ol>
Teaching methods
This course was held <em>live</em> in 2007, when it was recorded (audio/video). For subsequent years we make available the recordings for a remote teaching. The teacher receives students to answer questions and clarify doubts.<br />
<br />
The exam requires to solve a small number of difficult research-like problems, for which the students have one week. These problems are then discussed during an oral interview.
Assessment methods and criteria
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Other information
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