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On the area of excursion sets of spherical Gaussian eigenfunctions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2011 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, no 9, 093301- p.Article in journal (Refereed) Published
Abstract [en]

The high frequency behaviour for random eigenfunctions of the spherical Laplacian has been recently the object of considerable interest, also because of strong motivation arising from physics and cosmology. In this paper, we are concerned with the high frequency behaviour of excursion sets; in particular, we establish a uniform central limit theorem for the empirical measure, i.e., the proportion of spherical surface, where spherical Gaussian eigenfunctions lie below a level z. Our proofs borrow some techniques from the literature on stationary long memory processes; in particular, we expand the empirical measure into Hermite polynomials, and establish a uniform weak reduction principle, entailing that the asymptotic behaviour is asymptotically dominated by a single term in the expansion. As a result, we establish a functional central limit theorem; the limiting process is fully degenerate. (C) 2011 American Institute of Physics. [doi:10.1063/1.3624746]

Place, publisher, year, edition, pages
2011. Vol. 52, no 9, 093301- p.
Keyword [en]
eigenvalues and eigenfunctions, Gaussian processes, polynomials, random processes
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Other Physics Topics
Identifiers
URN: urn:nbn:se:kth:diva-50325DOI: 10.1063/1.3624746ISI: 000295622100018Scopus ID: 2-s2.0-80052004042OAI: oai:DiVA.org:kth-50325DiVA: diva2:480331
Note
Qc 20120119Available from: 2012-01-19 Created: 2011-12-05 Last updated: 2017-12-08Bibliographically approved

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