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A limit theorem for quadratic fluctuations in symmetric simple exclusion

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Assing, Sigurd (2007) A limit theorem for quadratic fluctuations in symmetric simple exclusion. Stochastic Processes and their Applications, Vol.117 (No.6). pp. 766-790. doi:10.1016/j.spa.2006.10.005 ISSN 0304-4149.

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Official URL: http://dx.doi.org/10.1016/j.spa.2006.10.005

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Abstract

We consider quadratic fluctuations V-epsilon(H) (eta(s)) = root epsilon Sigma(x is an element of Z) H(epsilon x)eta(s)(x)eta(s) (x + x(0)) in the centered symmetric simple exclusion process in dimension d = 1. Although the order of divergence of root E[(epsilon-2)(0) ds v(epsilon)(h) (eta(s))](2) is known to be epsilon(-3/2) if epsilon down arrow 0 the corresponding limit theorem was so far not explored. We now show that epsilon(3/2) integral(t epsilon)(0)(-2) ds V-epsilon(H) (eta(s)) converges in law to a non-Gaussian singular functional of an infinite-dimensional Ornstein-Uhlenbeck process. Despite the singularity of the limiting functional we find enough structure to conclude that it is continuous but not a martingale in t. We remark that in symmetric exclusion in dimensions d >= 3 the corresponding functional central limit theorem is known to produce Gaussian martingales in t. The case d = 2 remains open. (C) 2006 Elsevier B.V. All rights reserved.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Statistics
Journal or Publication Title: Stochastic Processes and their Applications
Publisher: Elsevier Science BV
ISSN: 0304-4149
Official Date: June 2007
Dates:
DateEvent
June 2007Published
Volume: Vol.117
Number: No.6
Number of Pages: 25
Page Range: pp. 766-790
DOI: 10.1016/j.spa.2006.10.005
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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