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Derived from
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20410345 CP410 - Theory of Probability in Computational Sciences LM-40 CAPUTO PIETRO
(syllabus)
Introductory example: the branching process.
Measure theory. Existence and uniqueness theorems for probability measures.
Borel-Cantelli lemma 1. Random variables. Independence.
Borel-Cantelli lemma 2. Kolmogorv's 0/1 law.
Integration. Expected value.
Monotone convergence and the dominated convergence theorem.
Inequalities: Markov, Jensen, Hoelder, Cauchy-Schwarz.
Laws of large numbers.
Product measures. Fubini's theorem. Joint laws.
Conditional expectation with respect to a sub sigma-algebra.
Martingales. Stopping times. Optional stopping and applications.
Hotting times. Convergence theorem for martingales bounded in L^1 and L^2.
Examples, Kolmogorov's strong law of large numbers.
Convergence in distribution and the central limit theorem.
(reference books)
D. Williams, Probability with martingales. Cambridge University Press, (1991).
R. Durrett, Probability: Theory and Examples. Thomson, (2000).
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Università Roma Tre