Instructor: Dr. Yuri Bakhtin
- (2006-12-08, NEW) Here is the list of what will be tested on the final exam.
- Conditional expectation and conditional probability as Radon--Nykodim derivatives. Equivalent definitions and basic properties (with proofs).
Regular conditional distributions, their existence (no proof). Conditional density.
- Conditional independence. Markov property: equivalence of various definitions.
Transition probability and density for Markov processes. Chapman--Kolmogorov equation. Finite-dimensional distributions of Markov processes.
Stopping times. Sigma-algebras associated to stopping times. Strong Markov property for discrete time Markov processes (with proofs)
- (Sub-, super-) martingales. Doob's decomposition, compensators. Optional sampling theorem. Maximal inequalities. Intersecion lemma and Convergence theorem (with proofs).
- Everything that was covered on the midterm tests, see below.
- Problems (MOSTLY similar to the homework problems or taken from the homework assignments)
- (2006-11-29) I just posted the last homework assignment,pp.5--6 It is due on Wednesday, December, 6. As usual, please report misprints!
- (2006-11-03) Here is the list of what will be on the test:
- CLT under Lindeberg's condition, particular cases: i.i.d., Lyapunov's condition, bounded r.v's (with proof).
- Lattice r.v.'s. Gnedenko's Local CLT (with proof).
- CLT for m-dependent r.v.'s: Bernstein's method (with proof).
- Berry-Esseen inequality (no proof).
- Properties of infinitely divisible distributions including their description as limit distributions (with proofs).
- Levy-Khinchin representation (no proof).
- Characterization of stable distributions as limit distributions for sums (no proof).
- Law of the Iterated Logarithm (proof for i.i.d. gaussian r.v.'s).
- Kolmogorov's 0-1 Law on tail events; Hewitt--Savage 0-1 Law on permutable events; 0-1 Law for time-shift-invariant events for
i.i.d. r.v.'s (ergodic property). (with proofs)
- Random walk. Criterion of recurrence for the origin (with proof).
- Problems (similar to the homework problems or taken from the homework assignments)
- (2006-10-27) I just posted a new set of problems (p.4).
This is due on Friday, Nov 3, so that I will return your papers on Monday, Nov 6, before the Test on Wednesday, Nov 8.
- (2006-10-11) Another misprint corrected in Problem 1.
- (2006-10-10) I corrected an inessential misprint in problem 4 of the third homework assignment.
- (2006-10-09) The third homework assignment is posted (p.3)
It is due by Wednesday, October 18. Please report all the misprints.
- (2006-09-20) Here is the list of what will be on the test:
- Basic properties of characteristic functions (= ch.f.'s) (with proofs).
- Connection between moments of a r.v. and Taylor expansion of ch.f. at 0. (with proofs)
- Uniqueness property of ch.f.'s and the inversion formula. (with proofs)
- Notions of sequential compactness and tightness. Prokhorov's theorem (no proof).
- Convergence theorem for ch.f.'s. (with proof)
- Problems (similar to the homework problems or taken from the homework assignments)
- (2006-09-13) A misprint was corrected in Problem 8 of the second assignment of the
homework. I replaced "i.i.d" by "independent".
- (2006-09-12) I have just posted updated homework problems with the second assignment. The new assignment is on page 2. It is due by Friday September 22. Please report all misprints to me!
- (2006-08-31) The first homework assignment is posted here .
It is due by Monday September 11th.
- (2006-08-24) The first homework assignment will appear online by Friday September 1.
and it will be due by Monday September 11th.
- (2006-08-24) All updates on the course will be posted on this web page.