[Matematisk statistik] [Matematikcentrum] [Lunds universitet]

NAMS003 (5 hp): PhD course on History-Dependent Processes

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.

Current information (last updated: 13 January 2016):

Lectures: Thursday 1315-1500
Room: MH:228 except on 21 January (MH:227)


Pemantle's survey


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 similar.
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.

Not covered: Doob inequalities. Stochastic approximations. Birth and death processes review. Embedding of urn in B&D processes.
Edge-reinforced random walk as random walk in random environment and open problems. Simple harmonic urn.


Please contact the lecturer...

Lecturer for Fall 2015: Stanislav Volkov