Rates of convergence for lamplighter processes

Häggstr?:om, Olle and Johan Jonasson

Stochastic Processes and their Applications, 1997, vol. 67, issue 2, 227-249

Abstract: Consider a graph, G, for which the vertices can have two modes, 0 or 1. Suppose that a particle moves around on G according to a discrete time Markov chain with the following rules. With (strictly positive) probabilities pm, pc and pr it moves to a randomly chosen neighbour, changes the mode of the vertex it is at or just stands still, respectively. We call such a random process a (pm, pc, pr)-lamplighter process on G. Assume that the process starts with the particle in a fixed position and with all vertices having mode 0. The convergence rate to stationarity in terms of the total variation norm is studied for the special cases with G = KN, the complete graph with N vertices, and when G = mod N. In the former case we prove that as N --> [infinity], ((2pc + pm)/4pcpm)N log N is a threshold for the convergence rate. In the latter case we show that the convergence rate is asymptotically determined by the cover time CN in that the total variation norm after aN2 steps is given by P(CN > aN2). The limit of this probability can in turn be calculated by considering a Brownian motion with two absorbing barriers. In particular, this means that there is no threshold for this case.

Date: 1997
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304-4149(97)00007-0
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:67:y:1997:i:2:p:227-249

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().