14448 - PROBABILITY II
Third Year. Second semester.
8 ECTS credits.
- Introduction: Laws of Large Numbers and Central Limit
Theorem, in their simplest forms. Easy proofs of both Weak and Strong
Laws are presented, with their meaning as related to the Central Limit
- Random variables and random vectors, in the language
of Measure Theory. An example - based presentation, meant to recall
the elementary Probability Calculus (from Prob.I, 2nd year), and to boost
it with the language and ideas of Measure Theory (3rd year, 1st semester).
- The general setting of Independence, and some
consequences. Uses of the Weak and Strong Laws. Including the Borel-Cantelli
lemmas, Kolmogorov's 0-1 Law and Wald's Lemma. Conditional expectation.Examples.
- Generating functions, and some of their applications.
Probability generating function. Moment generating function. Examples.
- Characteristic functions. Inversion Theorems and
some limit theorems (weak law, Poisson, Central Limit Theorem).
- Convergence of random variables. Almost sure convergence,
convergece in probability, p-mean convergence. Relationships.
- Extensions of the Weak Law. Applications: "St.Petersburg
paradox", Weierstrass approximation theorem.
- Strong Laws (Kolmogorov's theorems). Kolmogorov's
maximal inequality. Distinct versions of the strong law. Applications:
- Central Limit Theorem. Convergence in distribution
and the continuity theorem. Relationships with other notions of convergence.
The central limit theorem. Weaker conditions for the central limit theorem:
Lyapunov and Lindeberg conditions.
- Adams & Guillemin. Measure theory
and probability. Birkhauser, 1996.
- Breiman. Probability. Addison-Wesley, 1968.
- Durrett. Probability: theory and examples.
Wadsworth & Brooks/Cole, 1991.
- Feller; An introduction to probability theory
and its applications. (2nd ed.): Wiley, 1971.
- Grimmett & Stirzaker. Probability and random
processes. Oxford, 1992.
- Grimmett & Welsh. Probability an introduction.
- Taylor. An introduction to measure and
probability. Springer, 1997.