[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)

Texts:

Pemantle's survey

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 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. 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 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.

Examination: