[Matematisk statistik]
[Matematikcentrum]
[Lunds universitet]
Official syllabus: English, Swedish,
...and and wild horses
thundered through the sky taking fresh supplies
of reinforced railings to the Uncertain Areas.
- The Hitch Hikers Guide to the Galaxy, Chapter IX. |
Last lectures 8 will be after Christmas break, 14 January, same time and place.
Institution/Avdelning: Matematikcentrum, Matematisk Statistik
Lectures: | Thursday 13^{15}-15^{00} | |
Room: | MH:228 | except on 21 January (MH:227) |
Texts:
Pemantle's survey
Planned topics yet to cover:
Preliminary schedule:
Lectures |
Subject |
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 Z^{1} (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
similar. Rubin's construction for weighted urns. Explosion time is continuous. Super-linear ERRW on Z^{1}. |
7 |
Decoupling of [[a,0],[0,b]] and [[0,a],[a,0]] urns. OK Corral. Exchangeability and Pólya urn. |
8 |
Proof P(|R|=5)>0 and P(|R|<∞)=1 for VRRW on Z^{1} as in V01. |
Lecturer for Fall 2015: Stanislav Volkov