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Central limit theorems for the spectra of classes of random fractals
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Charmoy, Philippe H. A., Croydon, David A. and Hambly, Ben M. (2017) Central limit theorems for the spectra of classes of random fractals. Transactions of the American Mathematical Society, 369 . 8967-9013 . doi:10.1090/tran/7147
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Official URL: https://doi.org/10.1090/tran/7147
Abstract
We discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal, the continuum random tree. In the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit. We will show examples from a class of random fractals generated from Dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem. The Brownian continuum random tree can also be viewed as a random fractal generated by a Dirichlet distribution. The first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this, though the positivity of the variance arising in the central limit theorem is left open. In both cases these fractals can be described through a general Crump-Mode-Jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics. Our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known.
| Item Type: | Journal Article | ||||||
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| Alternative Title: | |||||||
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||
| Library of Congress Subject Headings (LCSH): | Fractals., Spectral sequences (Mathematics), Random walks (Mathematics) | ||||||
| Journal or Publication Title: | Transactions of the American Mathematical Society | ||||||
| Publisher: | American Mathematical Society | ||||||
| ISSN: | 0002-9947 | ||||||
| Official Date: | May 2017 | ||||||
| Dates: |
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| Volume: | 369 | ||||||
| Page Range: | 8967-9013 | ||||||
| DOI: | 10.1090/tran/7147 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Funder: | Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung [Swiss National Science Foundation] (SNSF), Berrow Foundation Scholarships | ||||||
| Related URLs: |
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