University of Oulu

Feng, De-Jun; Järvenpää, Esa; Järvenpää, Maarit; Suomala, Ville. Dimensions of random covering sets in Riemann manifolds. Ann. Probab. 46 (2018), no. 3, 1542--1596. doi:10.1214/17-AOP1210. https://projecteuclid.org/euclid.aop/1523520024

Dimensions of random covering sets in Riemann manifolds

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Author: Feng, De-Jun1; Järvenpää, Esa2; Järvenpää, Maarit2;
Organizations: 1Chinese University of Hong Kong
2University of Oulu
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 0.4 MB)
Persistent link: http://urn.fi/urn:nbn:fi-fe201804176586
Language: English
Published: Institute of Mathematical Statistics, 2018
Publish Date: 2018-04-17
Description:

Abstract

Let M, N and K be d-dimensional Riemann manifolds. Assume that A :=(An)n∈N is a sequence of Lebesgue measurable subsets of M satisfying a necessary density condition and x := (xn)n∈N is a sequence of independent random variables, which are distributed on K according to a measure, which is not purely singular with respect to the Riemann volume. We give a formula for the almost sure value of the Hausdorff dimension of random covering sets E(x, A) := lim supn→∞An(xn) ⊂ N. Here, An(xn) is a diffeomorphic image of An depending on xn. We also verify that the packing dimensions of E(x, A) equal d almost surely.

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Series: Annals of probability
ISSN: 0091-1798
ISSN-E: 2168-894X
ISSN-L: 0091-1798
Volume: 46
Issue: 3
Pages: 1542 - 1596
DOI: 10.1214/17-AOP1210
OADOI: https://oadoi.org/10.1214/17-AOP1210
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Subjects:
Copyright information: © Institute of Mathematical Statistics, 2018. Published in this repository with the kind permission of the publisher.