Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T17:23:10.415Z Has data issue: false hasContentIssue false
  • English
  • Français

The Correct Asymptotic Variance for the Sample Mean of a Homogeneous Poisson Marked Point Process

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

William Garner  and
Dimitris N. Politis
William Garner*
Affiliation:
University of California at San Diego
Dimitris N. Politis*
Affiliation:
University of California at San Diego
*
∗ Postal address: Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA.
∗ Postal address: Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The asymptotic variance of the sample mean of a homogeneous Poisson marked point process has been studied in the literature, but confusion has arisen as to the correct expression due to some technical intricacies. This note sets the record straight with regards to the variance of the sample mean. In addition, a central limit theorem in the general d-dimensional case is also established.

Keywords

Central limit theoremmarked point process

MSC classification

Primary: 60F05: Central limit and other weak theorems 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 889 - 892
Copyright
© Applied Probability Trust 

References

Ballani, F., Kabluchko, Z. and Schlather, M. (2012). Random marked sets. Adv. Appl. Prob. 44, 603–616.Google Scholar
Brillinger, D. R. (1973). Estimation of the mean of a stationary time series by sampling. J. Appl. Prob. 10, 419–431.CrossRefGoogle Scholar
Ivanov, A. V. and Leonenko, N. N. (1986). Statistical Analysis of Random Fields. Kluwer Publishers, Dordrecht.Google Scholar
Karr, A. F. (1986). Inference for stationary random fields given Poisson samples. Adv. Appl. Prob. 18, 406–422.Google Scholar
Karr, A. F. (1991). Point Processes and Their Statistical Inference, 2nd edn. Marcel Dekker, New York.Google Scholar
Kutoyants, Y. A. (1984a). On nonparametric estimation of intensity function of inhomogeneous Poisson process. Problems Control Inf. Theory 13, 253–258.Google Scholar
Kutoyants, Y. A. (1984b). Parameter Estimation for Stochastic Processes (Res. Exposition Math. 6). Heldermann, Berlin.Google Scholar
Masry, E. (1983). Nonparametric covariance estimation from irregularly-spaced data. Adv. Appl. Prob. 15, 113–132.CrossRefGoogle Scholar
Politis, D. N., Paparoditis, E. and Romano, J. P. (1999). Resampling marked point processes. In Multivariate Analysis, Design of Experiments, and Survey Sampling (Statist. Textbooks Monogr. 159), ed. Ghosh, Subir, Marcel Dekker, New York, pp. 163–185.Google Scholar