On the rate of convergence in the central limit theorem for hierarchical Laplacians
ESAIM: Probability and Statistics, Tome 23 (2019), pp. 68-81.

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 = L$$. Choosing a sequence {ε(B)} of i.i.d. random variables we define the perturbed function C(B, ω) and the perturbed hierarchical Laplacian L$$ = L$$. We study the arithmetic means $$ of the L$$-eigenvalues. Under certain assumptions the normalized arithmetic means $$ 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.

Reçu le :
Accepté le :
DOI : 10.1051/ps/2018010
Classification : 12H25, 60F05, 94A17, 47S10, 60J25
Mots-clés : Ultrametric space, p-adic numbers, hierarchical Laplacian, fractional derivative, total variation and entropy distance
Bendikov, Alexander 1 ; Cygan, Wojciech 1

1 
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     title = {On the rate of convergence in the central limit theorem for hierarchical {Laplacians}},
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Bendikov, Alexander; Cygan, Wojciech. On the rate of convergence in the central limit theorem for hierarchical Laplacians. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 68-81. doi : 10.1051/ps/2018010. http://archive.numdam.org/articles/10.1051/ps/2018010/

[1] M. Aizenman and S.A. Molchanov, Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157 (1993) 245–278. | DOI | MR | Zbl

[2] S. Albeverio and W. Karwowski, A random walk on p-adic numbers: generator and its spectrum. Stoch. Process. Appl. 53 (1994) 1–22. | DOI | MR | Zbl

[3] A. Bendikov, W. Cygan and W. Woess, Oscillating heat kernels on ultrametric spaces. J. Spectr. Theory 9 (2019) 195–226. | DOI | MR | Zbl

[4] A. Bendikov, A. Grigoryan, S. Molchanov and G. Samorodnitsky, On a class of random perturbations of the hierarchical Laplacian. Izv. Math. 79 (2015) 859–893. | DOI | MR | Zbl

[5] A. Bendikov, A. Grigoryan and C. Pittet, On a class of Markov semigroups on discrete ultrametric spaces. Potential Anal. 37 (2012) 125–169. | DOI | MR | Zbl

[6] A. Bendikov, A. Grigoryan, C. Pittet and W. Woess, Isotropic Markov semigroups on ultrametric spaces. Russ. Math. Surv. 69 (2014) 589–680. | DOI | MR | Zbl

[7] A. Bendikov and P. Krupski, On the spectrum of the hierarchical Laplacian. Potential Anal. 41 (2014) 1247–1266. | DOI | MR | Zbl

[8] S.G. Bobkov, G.P. Chistyakov and F. Götze, Berry- Esseen bounds in the entropic central limit theorem. Probab. Theory Relat. Fields 159 (2014) 435–478. | DOI | MR | Zbl

[9] M. Del Muto and A. Figà-Talamanca, Diffusion on locally compact ultrametric spaces. Expositiones Math. 22 (2004) 197–211. | DOI | MR | Zbl

[10] M. Del Muto and A. Figà-Talamanca, Anisotropic diffusion on totally disconnected abelian groups. Pac. J. Math. 225 (2006) 221–229. | DOI | MR | Zbl

[11] F.J. Dyson, Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12 (1969) 91–107. | DOI | MR | Zbl

[12] O. Johnson, Information Theory and the Central Limit Theorem. Imperial College Press (2004). | DOI | MR | Zbl

[13] A.N. Kochubei, Pseudo-Differential Equations and Stochastics Over Non-Archimedian Fields. Vol. 244 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc. (2001). | MR | Zbl

[14] E. Kritchevski, Hierarchical Anderson Model. Vol. 42 of CRM Proceedings and Lecture Notes. Centre de Recherches Mathématiques (2007). | DOI | MR | Zbl

[15] E. Kritchevski, Spectral localization in the hierarchical Anderson model. Proc. Am. Math. Soc. 135  (2007) 1431–1440. | DOI | MR | Zbl

[16] E. Kritchevski, Poisson Statistics of Eigenvalues in the Hierarchical Anderson Model. Annales Henri Poincaré 9 (2008) 685–709. | DOI | MR | Zbl

[17] S. A. Molchanov, Hierarchical andom matrices and operators, application to Anderson model, in Proceedings of 6th Lucacs Symposium (1996) 179–194. | MR | Zbl

[18] M. S. Pinsker, Information and Information Stability of Random Variables and Processes. Holden-Day (1964). | MR | Zbl

[19] A.N. Shiryaev, Probability, 2nd edn. Springer (1996). | DOI | MR

[20] M.H. Taibleson, Fourier Analysis on Local Fields. Princeton University Press (1975). | MR | Zbl

[21] A.W. Van Der Vaart, Asymptotic Statistics. Cambridge University Press (1998). | DOI | MR | Zbl

[22] V.S. Vladimirov, Generalized functions over the field of p-adic numbers. Uspekhi Matematicheskikh Nauk 43 (1988) 17–53. | MR | Zbl

[23] V.S. Vladimirov, I.V. Volovich and E.I. Zelenov, p-adic analysis and mathematical physics. Vol. 1 of Series on Soviet and East European Mathematics. World Scientific Publishing Co. Inc. River Edge (1994). | MR | Zbl

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