Mathematics BSc
Course description
2013.
Borromean Rings
Probability Theory 2
Hours
lect+pc		 Credits
lect+pc		 Assessment Specialization Course code
lect/pc		 Semester Status
3 + 2 3 + 2 exam +
term grade		 pure math. mm1c1vs5m
mm1c2vs5m		 5 alternative
Strong Weak Prerequisites
Practice class
Strong:
Probability theory1L-m (mm1c1vs3m) or
Probability theory1L-a (mm1c1vs3a)
Strong:
Analysis4L-p (mm1c1an4m) or
Analysis4L-a (mm1c1an4a)
Lecture
Weak:
practice class
Remarks
  • Both pure and applied mathematics students can choose whether to take the applied or pure mathematics version of Probability theory 2 (it is compulsory to take one of these two courses). The pure mathematics version covers more material.
Syllabus designed by:
Literature
    Recommended:
    • A. A. Borovkov: Probability Theory. Springer, 2013.
    • W. Feller: An Introduction to Probability Theory and Its Applications. Vol. 2, 2nd edition, Wiley, 1971.
    • L. Galambos: Advanced probability theory. Marcel Dekker, New York, 1995.
    • A. N. Shiryaev: Probability. 2nd edition, Springer, 1995.
    Syllabus
    • Kolmogorov’s probability field. Random variables, distribution, cumulative distribution function, density function. Expectation, variance.
    • Independence of event, random variables. Independence of the generated sigma-fields. Kolmogorov’s zero-one law.
    • Convergence in probability, almost sure, in Lp. Uniform integrability. De la Vallée Poussin’s theorem.
    • Lévy’s inequality. Equivalence of the convergence in probability and almost sure convergence for the sums of independent random variables. Weak laws of large numbers. Feller’s WLLN.
    • Weak convergence of random variables. Tightness, relative compactness. Prohorov’s theorem.
    • Characteristic functions of random variables. Inversion formulae. Unicity theorem. Continuity theorem. Doob’s inequality. Central limit theorem. Lindeberg-Feller’s theorem. Rate of convergence (Berry-Esséen’s theorem).
    • Conditional expectation with respect to sigma-fields. Conditional density function. Regular version of the conditional distribution.
    • Martingales, maximal-inequalities. almost sure convergence of martingales.
    • Strong law of large numbers. Kolmogorov’s theorem for independent, identically distributed random variables.
    • Almost sure convergence of sums of independent random variables. Kolmogorov’s three series theorem.
    • L1 convergence of martingales. Regular martingales.
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