
Stochastik II – Stochastics IISummer 2015 


In a nutshell...
Starting: 14th April 2015
Place: Arnimallee 6
Lectures: Tuesday (SR 007/008) and Thursday (SR 025/026), 08:0010:00
Problems: Wednesday (SR 025/026) 08:0010:00 (Time and place changed!!!)
More fun: I will be available in my office at Tuesday 10:0012:00. Of course meetings can be also arranged by appointment.
Prerequisites: a previous course on probability theory is required. Additionally, courses in analysis will be also desirable.
Synopsis: this
course is the sequel of the course of Stochastik I. The main
objective is to go beyond the first principles in probability theory
by introducing the general language of measure theory, and the
application of this framework in a wide variety of probabilistic
scenarios. More precisely, the course will cover the following
aspects of probability theory:
 Measure theory and the
Lebesgue integral.
 Convergence of random variables and 01
laws.
 Generating functions: Branching processes and
characteristic functions.
 Markov chains
 Introduction to
martingales
Feel free to contact me at jrue at zedat.fuberlin.de with any questions prior to the start of the course!
Requirements: the requirements of the course are here.
What we have seen so far...
When? 
What? 
Further Reading 
Problems 
14/04/15 
Introduction: from Riemann to Lebesgue. Characteristic functions and step functions. Riemann integral. Motivation on why is needed to extend the notion of Riemann integral. Sigmaalgebras and measurable functions: definition of sigmaalgebra, first observations and examples. Sigmaalgebra generated by a set and Borel sets. Measurable functions: definition , equivalence of properties and first examples (constant functions, characteristic functions and continous functions). 
Sigmaalgebras, functions and measures
Important: Wednesdays session change to 08:0010:00 (SR 025/026) 

15/04/15 
Measurable functions: sum, product, absolute value and others are measurable (trick using rational numbers for proving measurability of the sum). Measurability in the extended line. Measurability of inf, sup, lim inf and lim sup. Product of functions in the extended line. How to build a monotne increasing sequence of functions that converges pontwise to a given function. 

16/04/15 
Measurable functions between (general) measurable spaces. Measure over a measurable space: definition, and properties. First examples and relation with monotone sequences of measurable sets. Lebesgue measure over R: algebras and measures over algebras. The algebra generated by the intervals. 

21/04/15 
Lebesgue measure over R: outer measure, Caratheodory's condition and Caratheodory's Theorem (No proof). Lebesgue sets and Lebesgue measure over Borel sets. Integral of positive functions: simple functions and canonical representation. Integral of a simple function and first consequences. Integral of positive functions and first consequences. Statement of the Monotone Convergent Theorem. 
Integral of positive functions, and the Monotone Convergence Theorem 

22/04/15 
Integral of positive functions: proof of the Monotone Convergence Theorem. Consequences: linearity of the integral, Fatou Lemma, construction of measures from integrable functions. Functions that are 0 mua.a. And extension of MCT to cover such functions. 

23/04/15 
Lecture Cancelled 

28/04/15 
Integrals over general functions: definitions and integral of the absolute value. The Dominated Convergence Theorem. Applications: integrals depending on parameters. Limits with respect to parameters: continuity. 
Dominated Convergence Theorem, parameters and extranotions in measure thory 

29/04/15 
Problem Session 

30/04/15 
Limits with respect to parameters: derivatives. Extra notions: the relation between the Riemann integral and the Lebesgue integral. Absolutely continous measures, RadonNikodym Theorem (No proof) and the RadonNikodym derivative. Product of measure spaces, existence and unicity of product measure. 

05/05/15 
Problem Session 

(correction in Probl 2) 
06/05/15 
Problem Session 

07/05/15 
Product
of measures: Fubini's Theorem 

12/05/15 
The ABC in Probability theory: Pprobability spaces, events and probability functions. Random variables: law of a random variable, density probability functions and probability distribution functions. Expectation of a random variable, rth moment and variance. Results for the expectation arising from measure theory. Independence of events and independence of random variables. Conditional probability and conditional expectation. 


13/05/15 
Problem Session 

14/05/15 
Holiday (NO lecture) 

19/05/15 
The ABC in Probability theory: vectors of random variables. Law of a vector of random variables, and restriction to the case of independent random variables. Modes of convergence for random variables: convergence almost surely, in probability, in rth mean mode and in distribution. Implications: convergence in probability implies convergence in distribution. Convergence in 1th mean mode implies convergence in probability. 


20/05/15 
Problem Session 

21/05/15 
Modes of convergence for random variables: implications between mean modes of convergence. Convergence almost surely implies convergence in probability: proof and related concepts. Inverse results (convergence in distribution + additional condition implies convergence in probability; convergence in probability + additional condition implies convergence in rth mean). Skorokhod' Representation Theorem and consequences. 

26/05/15 
Problem Session 

Typo corrected in Probl 4 
27/05/15 
Problem Session 

28/05/15 
Modes of convergence: proof of Skorokhod's Representation Theorem. 01 Laws: First and second BorelCantelli lemmas. Independence of families of events. Sigmaalgebra generated by a family of random variables. Tail sigmaalgebra and tail event. Statement of Kolmogorov's 01 Law. 

02/06/15 
Proof of Kolmogorov's 01 Law: family of independent families of events implies that the sigmaalgebras are also independent. Final arguments for the proof of Kolmogorov's 01 law. Probability generating functions of positive discrete random variables: definitions and first examples. Properties of formal power series as analytic obtects. First properties of probability generating functions (computation of factorial moments, convexity). 

(correction in Prob. 2) 
03/06/15 
Problem Session 

04/06/15 
Probability generating functions: sum of independent random variables and product of GFs. Sum of independent identically distributed random variables depending on a random index: composition of GFs. Branching processes: model and equations. Probability GF for the number of members in a given generation of the process. Expectation and variance of the number of members in a given generation. Extinction and probability of extinction. Three regimes (subcritical, critical, supercrticial) and behaviour. 

09/06/15 
Branching processes: probability of extinction: proofs for the three regimes. Extinction time: definition and relation with the probability generating function. Behaviour in the subcritical case (exponential decay) and the critical case (polinomial decay). Proof of the subcritical situation. 


10/06/15 
Problem Session 

11/06/15 
Branching processes: extinction time in the critical case (sketch). Random walks: definition of random walk. Escape from 0, recurrence and transience. Polya's transient theorem. Strategy: encoding by means of generating functions. Study of the unidimensional and bidimensional case (recurrent families). 

16/06/15 
Random walks: Transient case: dimension 3. Further comments further comments on higher dimensions, random walks with barriers and self avoiding walks. Moment GFs: definition, first properties related with the independence and sums, examples. 


17/06/15 
Moment GFs: bounds for tail estimates in terms of moment gf. Theorems for moment gfs: continuity and limits. Law of large numbers and Central Limit Theorem. Characteristic functions and Fourier transform. 

18/06/15 
Stochastic processes: definition, and defintion of Markov chain. equivalent formulations and timehomogeneous Markov chain. Examples: random walks, random walks with barrier, branching processes. Transition matrix. 

23/06/15 
Markov chains: ChapmanKolmogorov's relations, evolution of the probability distribution of the Markov chain in terms of the transition matrix. Classification of states: recurrent and transient states. Characterization. Mean recurrence time, null (and nonnull) states, period of an state, ergodic states and absorving states. 


24/06/15 
Problem Session 

25/06/15 
Markov chains: interaction between states: comunicacion and intercomunication. Closed and irreducible set of states. Decomposition Theorem for Markov chains. Stationary distributions: definitions and first properties. 

30/06/15 
Problem Session 


01/07/15 
Problem Session 

02/07/15 
Problem Session 

07/07/15 
Stationary distributions: construction of stationary distributions from the mean recurrence time. Lemma building an stationary distribution by using a nonnull recurrent state. Main theorem on irreducible nonnull recurrent Markov chains vs stationary distributions. Limit theorems in terms of the mean recurrence time. 


08/07/15 
Problem Session 

09/07/15 
Timereversible Markov chains: definition and characterization. Ehrenfest model for urns. Finite state Markov chains: PerronFrobenius theorem for stochastic matrices and applications (finite irreducible Markov chains are nonnull recurrent). 

14/07/15 
Martingales: the gambler ruin and definition of martingale. Properties of the conditional expectation. Examples of martingales. Convergence theorem for martingales. 


15/07/15 
Problem Session 

16/07/15 
Problem Session 

21/07/15 
FINAL EXAM: starts at 9:00 at Arnimallee 6, SR032 Solution to Problems 4 and 5 

15/09/15 
MAKEUP EXAM 15^{th} September: starts at 9:00 at Arnimallee 6, SR032 

18/09/15 