Official syllabus: English, Swedish,
Current information (last updated: 13 January 2016):
|Room:||MH:228|| except on 21 January (MH:227)
|1||Introduction, Markov chains and
their classification, methods for non-Markov processes,
Borel-Cantelli lemmas (incl Levy), Chernoff-Cramer theorem.
|2||Review of convergence
theorems and martingales, MCT, proof of Levy's BC lemma.
|3||MCT proof of (1) convergence of
Pólya urn; (2) that the limit is in the interior;
(3) convergence of irreducible GPU to left eigenvector of Q.
|4||Degenerate GPUs when abcd=0.
3-step martingale proof of Thm2.3 PV99.
VRRW definitions. Proof P(|R|<5)=0 on Z1 (done for cases |R|=1,2,3)
|5||Proof P(|R|=4)=0 using
GPU([[1,0],[c,1]]) as in PV99 - complete proof in 7 steps.
General qualitative properties of VRRW on various graphs from PV99, V2001 and Tarres2004.
|6|| Birth-death processes methods.
Solution to Pólya urn and more general urns using Yule process and
Rubin's construction for weighted urns. Explosion time is continuous. Super-linear ERRW on Z1.
||Decoupling of [[a,0],[0,b]] and [[0,a],[a,0]] urns. OK Corral. Exchangeability and Pólya urn.|
||Proof P(|R|=5)>0 and P(|R|<∞)=1 for VRRW on Z1 as in V01.|
Lecturer for Fall 2015: Stanislav Volkov