Dimensions of random covering sets in Riemann manifolds 

Author:  Feng, DeJun^{1}; Järvenpää, Esa^{2}; Järvenpää, Maarit^{2}; 
Organizations: 
^{1}Chinese University of Hong Kong ^{2}University 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:fife201804176586 
Language:  English 
Published: 
Institute of Mathematical Statistics,
2018

Publish Date:  20180417 
Description: 
AbstractLet 𝐌, 𝐍 and 𝐊 be ddimensional Riemann manifolds. Assume that 𝐀 := (A_{n})_{n∈ℕ} is a sequence of Lebesgue measurable subsets of 𝐌 satisfying a necessary density condition and 𝐱 := (x_{n})_{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 sup_{n→∞}A_{n}(x_{n}) ⊂ 𝐍. Here, A_{n}(x_{n}) is a diffeomorphic image of A_{n} depending on x_{n}. We also verify that the packing dimensions of 𝐄(𝐱, 𝐀) equal d almost surely. see all

Series: 
Annals of probability 
ISSN:  00911798 
ISSNE:  2168894X 
ISSNL:  00911798 
Volume:  46 
Issue:  3 
Pages:  1542  1596 
DOI:  10.1214/17AOP1210 
OADOI:  https://oadoi.org/10.1214/17AOP1210 
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. 