- Probability Theory 2
- Lecture:
- Monday 10-12, H46
-
- Practice:
- Wednesday 8-10, T606
-
-
General overview
This is a second course of introductory probability designed for mathematics BSc students. It is assumed that the attending students have completed a first course in probability covering standard material and are aware of basic notions like random variables and their distributions, conditional probability and stochastic independence, Bernoulli's Law of Large Numbers and De Moivre's Central Limit Theorem.
-
Main topics of the semester
- Sums of independent random variables and convolution of distributions (with introduction to Riemann-Stieltjes integral). Applications: Gaussian, Cauchy, exponential, Gamma, etc.
- The generating function. Applications: branching processes, hitting times and occupation times of random walks, weak convergence of discrete distributions and Poisson approximation, etc.
- The Weak Law of Large Numbers: Chebyshev's and Markov's inequalities and the WLLN. Applications.
- Tail and large deviation estimates for sums of independent random variables: Bernstein, Hoeffding, Chernoff and Cramér bounds applications.
- Convergence in probability and almost sure convergence. The Borel-Cantelli Lemma. Application: the Strong Law of Large Numbers assuming fourth moment.
- Kolmogorov's inequality and the Two Series Theorem, Kolmogorov's Strong Law of Large Numbers (in full detail). Kolmogorov's 0-1 Law.
- The Characteristic Function 1: definition; basic properties; moments of rv and derivatives of its chf; smoothness of pdf and decay of chf; inversion.
- Weak Convergence of Probability Distribution Functions: definition and characterizations; tightness and subsequential weak convergence.
- The Characteristic Function 2: pointwise convergence of chf-s and weak convergence of pdf-s. Application: The Central Limit Theorem in its full strength.
-
- Requirements
- Topics for the oral exam (On the exam the students present two topics, one is chosen by the student and one is given by the examiner.
-
Recommended books
- William Feller: An Introduction to Probability Theory vol 1-2, 1968-71.
- Alfréd Rényi: Probability Theory. 1970.
- Rick Durrett: Probability: theory and examples, 2010.
-
Handwritten notes of Bálint Tóth
- The convolution,
- The generating function,
- Concentration inequalities,
- Strong Laws,
- Characteristic function,
- Weak convergence,
- Supplementary: Measure and Integration I, Measure and integration II, Measure and integration III, Etemadi's proof of SLLN
-
- Consultation/office hour: Wednesday 10:15-11:00, room: H506
-
Samples for midterm:
- First midterm
- Second midterm