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 are equal to the upper and lower limit of , where is the mean distance to the closest point for the Poisson point processes with intensity measure . Moreover the upper local dimension of is a.e. bounded from above by the limit superior of , where 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

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.