Mathematics BSc
Course description
2013. Probability Theory 2
Hours
lect+pc
Credits
lect+pc
Assessment Specialization Course code
lect/pc
Semester Status
3 + 2 3 + 2 exam +
pure math. mm1c1vs5m
mm1c2vs5m
5 alternative
Strong Weak Prerequisites
Practice class
Strong:
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.