Probability Theory II* 1000-135RP2*
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.
Type of course
Course coordinators
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Learning outcomes
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.
Assessment criteria
Examination
Bibliography
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.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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