Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Hausdorff dimension of affine random covering sets in torus

Esa Järvenpää, Maarit Järvenpää, Henna Koivusalo, Bing Li, and Ville Suomala

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Abstract

We calculate the almost sure Hausdorff dimension of the random covering set $\limsup_{n\to\infty}(g_{n}+\xi_{n})$ in $d$-dimensional torus $\mathbb{T}^{d}$, where the sets $g_{n}\subset\mathbb{T}^{d}$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and $\xi_{n}\in\mathbb{T}^{d}$ are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.

Résumé

Nous calculons presque sûrement la dimension de Hausdorff de l’ensemble de recouvrement aléatoire $\limsup_{n\to\infty}(g_{n}+\xi_{n})$ dans le tore $\mathbb{T}^{d}$ de dimension $d$, où $g_{n}\subset\mathbb{T}^{d}$ sont des parallélépipèdes, ou plus généralement, des images linéaires d’un ensemble d’intérieur non vide et $\xi_{n}\in\mathbb{T}^{d}$ sont des points aléatoires indépendants et uniformément distribués. La formule de dimension, exprimée en fonction des valeurs singulières des applications linéaires, est valable à condition que la suite de ces valeurs singulières soit décroissante.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 4 (2014), 1371-1384.

Dates
First available in Project Euclid: 17 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1413555504

Digital Object Identifier
doi:10.1214/13-AIHP556

Mathematical Reviews number (MathSciNet)
MR3269998

Zentralblatt MATH identifier
1319.60016

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 28A80: Fractals [See also 37Fxx]

Keywords
Random covering set Hausdorff dimension Affine Cantor set

Citation

Järvenpää, Esa; Järvenpää, Maarit; Koivusalo, Henna; Li, Bing; Suomala, Ville. Hausdorff dimension of affine random covering sets in torus. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 4, 1371--1384. doi:10.1214/13-AIHP556. https://projecteuclid.org/euclid.aihp/1413555504


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