dc.contributor.author |
van der Hofstad, Remco |
en |
dc.contributor.author |
Holmes, Mark |
en |
dc.date.accessioned |
2012-05-03T21:42:24Z |
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dc.date.issued |
2012 |
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dc.identifier.citation |
Brazilian Journal of Probability and Statistics 26(1):1-55 01 Feb 2012 (Available online 2011) |
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dc.identifier.issn |
0103-0752 |
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dc.identifier.uri |
http://hdl.handle.net/2292/17760 |
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dc.description.abstract |
We derive a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the true (weakly) self-avoiding walk, loop-erased random walk, and annealed random walk in random environment. In this paper we show that the expansion gives rise to useful formulae for the speed and variance of the random walk, when these quantities are known to exist. The results and formulae of this paper have been used elsewhere by the authors to prove monotonicity properties for the speed (in high dimensions) of excited random walk and related models, and certain models of random walk in random environment. We also derive a law of large numbers and central limit theorem (with explicit error terms) directly from this expansion, under strong assumptions on the expansion coefficients. The assumptions are shown to be satisfied by excited random walk in high dimensions with small excitation parameter, a model of reinforced random walk with underlying drift and small reinforcement parameter, and certain models of random walk in random environment under strong ellipticity conditions. |
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dc.language |
English |
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dc.publisher |
Brazilian Statistical Association |
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dc.relation.ispartofseries |
Brazilian Journal of Probability and Statistics |
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dc.rights |
Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher. Details obtained from http://www.sherpa.ac.uk/romeo/issn/0103-0752/ |
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dc.rights.uri |
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm |
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dc.subject |
Science & Technology |
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dc.subject |
Physical Sciences |
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dc.subject |
Statistics & Probability |
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dc.subject |
Mathematics |
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dc.subject |
Self-interacting random walks |
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dc.subject |
excited random walk |
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dc.subject |
once-edge-reinforced random walk |
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dc.subject |
random walk in random environment |
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dc.subject |
lace expansion |
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dc.subject |
law of large numbers |
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dc.subject |
central limit theorem |
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dc.subject |
SUPER-BROWNIAN MOTION |
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dc.subject |
UPPER CRITICAL DIMENSION |
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dc.subject |
AVOIDING RANDOM-WALK |
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dc.subject |
EXCITED RANDOM-WALK |
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dc.subject |
CRITICAL ORIENTED PERCOLATION |
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dc.subject |
INCIPIENT INFINITE CLUSTER |
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dc.subject |
REINFORCED RANDOM-WALK |
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dc.subject |
SCALING LIMIT |
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dc.subject |
LATTICE TREES |
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dc.subject |
LACE EXPANSION |
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dc.title |
An expansion for self-interacting random walks |
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dc.type |
Journal Article |
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dc.identifier.doi |
10.1214/10-BJPS121 |
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pubs.issue |
1 |
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pubs.begin-page |
1 |
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pubs.volume |
26 |
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dc.rights.holder |
Copyright: Brazilian Statistical Association |
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pubs.end-page |
55 |
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dc.rights.accessrights |
http://purl.org/eprint/accessRights/RestrictedAccess |
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pubs.subtype |
Article |
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pubs.elements-id |
267537 |
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pubs.record-created-at-source-date |
2012-05-04 |
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