Probability Theory II

elective courses

(in Polish) Oczekuje się dobrej znajomości zagadnień ujętych w sylabusach przedmiotów Analiza matematyczna II.1 oraz Rachunek Prawdopodobieństwa I.

The course contains an introduction to the convergence theory for probability distributions (equivalence of various definitions, Central Limit Theorem) and to applications of harmonical analysis in the theory (properties of characteristic functions). Moreover, some elements of martingale theory and Markov chains will be discussed.

Convergence of probability distributions. The characteristic function of a probability distribution, applications to computing moments and distributions of sums of independent random variables. The uniqueness theorem. Levy's theorem stating that the convergence of probability distributions can be described in terms of the pointwise convergence of their characteristic functions. The Central Limit Theorem. Introduction to the theory of martingales ("fair games"). Stopping moments. Doob's "optional sampling" theorem. Markov chains, ergodicity.

1. A. N. Shiryaev, ,,Probability'', 2nd Edition, Spriger Verlag, 1989.

2. P. Billingsley, ,,Probability and Measure'', 3rd Edition, John Wiley & Sons, Inc., 1995.

3. Kai Lai Chung, ,,A Course in Probability Theory'', Revised 2nd Edition, Academic Press, 2001.

A student

1. understands the concept of convergence in law and its various characterizations (for example: in terms of pointwise convergence of density functions, atoms, distribution functions). A student knows the definition of tightness and Prokhorov's theorem.;

2. understands the concept of characteristic function of a probability distribution. He/She knows how to deduce various properties of a probability distribution from its characteristic function. He/She knows how the convergence in law relates to pointwise convergence of characteristic functions.

3. knows Central Limit Theorem in a general form (with the Lindeberg condition). He/She can give examples of its usefulness in applications.

4. understands the concept of conditional expectation and its properties; he/she can apply this concept to solve the problem of prediction;

5. knows the concepts of filtration and stopping time;

6. knows the concept of discrete time martingale, supermartingale and submartingale and basic inequalities related to these processes. He/she Knows conditions for almost sure convergence of such processes. He/she can characterize the convergence of martingales in L_p;

7. knows the concept of a Markov chain and related objects (state space, transition matrix, initial distribution, stationary distribution, etc.). He/she can

classify the states (periodic, recurring, momentary). He/she knows the ergodic theorem and its applications.