Probability Theory II*

elective courses

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

The course is intended for students interested in a deeper understanding of probability theory and willing to think about various related problems

and excercises.

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.

Usually this course follows closely the book of Jakubowski and Sztencel (in Polish). Most of it, in a similar, though not identical, exposition can be found in the classical books "Probability and measure" and "Convergence of probability measures" by Patrick Billingsley and "An introduction to probability theory and its applications" by William Feller.

A student

1. knows the definition of the convergence in distribution and its various characterizations (in terms of convergence of cummulative distribution functions etc.), as well as the definition of tightness and Prokhorov's theorem;

2. knows the definition of the characteristic function of a random variable and is able to deduce various properties of a probability distribtion from its characteristic function; can express the convergence in distribution in terms of the pointwise convergence of characteristic functions;

3. knows the Central Limit Theorem (under the Lindeberg condition assumptions) and its applications; knows the Berry-Esseen theorem;

4. knows the definition of a multidimensional Gaussian distribution and knows its characteristic function; knows that uncorrelated coordinates of a Gaussian vector are independent; is able to formulate the multidimensional Central Limit Theorem

5. knows the definition of a martingale, supermartingale and submartingale (with discrete time) and basic inequlities related to these processes; knows conditions that imply the almost sure convergence of these processes; knows the definition of uniform convergence and characterization of convergence of martingale in L_p;

6. knows the definition of a Markov chain and related objects (state space, transition matrix, initial distribution, stationary distribution, etc.); knows the classification of states (periodic, recurrent, transient) and recurrence criteria, as well as the ergodic theorem and its applications.

Examination