| Issue |
ESAIM: PS
Volume 23, 2019
|
|
|---|---|---|
| Page(s) | 68 - 81 | |
| DOI | https://doi.org/10.1051/ps/2018010 | |
| Published online | 14 March 2019 | |
On the rate of convergence in the central limit theorem for hierarchical Laplacians★
1
Instytut Matematyczny, Uniwersytet Wrocławski,
Pl. Grunwaldzki 2/4,
50-384
Wrocław, Poland.
2
Institut für Diskrete Mathematik, Technische Universität Graz,
Steyrergasse 30,
8010
Graz, Austria.
* Corresponding author: woj.cygan@gmail.com
Received:
24
February
2017
Accepted:
13
March
2018
Let (X, d) be a proper ultrametric space. Given a measure m on X and a function B↦C(B) defined on the set of all non-singleton balls B we consider the hierarchical Laplacian L = LC. Choosing a sequence {ε(B)} of i.i.d. random variables we define the perturbed function C(B, ω) and the perturbed hierarchical Laplacian Lω = LC(ω). We study the arithmetic means λ̅(ω) of the Lω-eigenvalues. Under certain assumptions the normalized arithmetic means (λ̅−Eλ̅) ∕ σ(λ̅) converge in law to the standard normal distribution. In this note we study convergence in the total variation distance and estimate the rate of convergence.
Mathematics Subject Classification: 12H25 / 60F05 / 94A17 / 47S10 / 60J25
Key words: Ultrametric space / p-adic numbers / hierarchical Laplacian / fractional derivative / total variation and entropy distance
© EDP Sciences, SMAI 2019
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