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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nearest neighbour distance and dimension of intensity measure of Poisson point process


Author: Radosław Wieczorek
Journal: Proc. Amer. Math. Soc. 139 (2011), 139-152
MSC (2010): Primary 28A80; Secondary 60G55
Published electronically: June 29, 2010
MathSciNet review: 2729078
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Abstract: We prove that the upper and lower local dimensions of a finite measure $ \mu$ are equal to the upper and lower limit of $ -\log \alpha / \log \hat{r}_{\alpha \mu}(x)$, where $ \hat{r}_{\alpha \mu}(x)$ is the mean distance to the closest point for the Poisson point processes with intensity measure $ \alpha \mu$. Moreover the upper local dimension of $ \mu$ is a.e. bounded from above by the limit superior of $ -\log \alpha / \log \hat{r}_{\alpha \mu}$, where $ \hat{r}_{\alpha \mu}$ denotes the expected nearest-neighbour distance.


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Additional Information

Radosław Wieczorek
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Bankowa 14, 40-007 Katowice, Poland
Email: r.wieczorek@impan.gov.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10467-7
PII: S 0002-9939(2010)10467-7
Keywords: Pointwise dimension, nearest-neighbour distance, measure, Poisson point processes
Received by editor(s): October 5, 2009
Received by editor(s) in revised form: November 6, 2009, February 6, 2010, and February 18, 2010
Published electronically: June 29, 2010
Additional Notes: This research was partially supported by the State Committee for Scientific Research (Poland) Grant No. N N201 0211 33 and by EC FP6 Marie Curie ToK programme SPADE2, MTKD-CT-2004-014508 and Polish MNiSW SPB-M
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.