The Annals of Statistics

Stable marked point processes

Tucker McElroy and Dimitris N. Politis

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Abstract

In many contexts such as queuing theory, spatial statistics, geostatistics and meteorology, data are observed at irregular spatial positions. One model of this situation involves considering the observation points as generated by a Poisson process. Under this assumption, we study the limit behavior of the partial sums of the marked point process {(ti, X(ti))}, where X(t) is a stationary random field and the points ti are generated from an independent Poisson random measure ℕ on ℝd. We define the sample mean and sample variance statistics and determine their joint asymptotic behavior in a heavy-tailed setting, thus extending some finite variance results of Karr [Adv. in Appl. Probab. 18 (1986) 406–422]. New results on subsampling in the context of a marked point process are also presented, with the application of forming a confidence interval for the unknown mean under an unknown degree of heavy tails.

Article information

Source
Ann. Statist., Volume 35, Number 1 (2007), 393-419.

Dates
First available in Project Euclid: 6 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1181100192

Digital Object Identifier
doi:10.1214/009053606000001163

Mathematical Reviews number (MathSciNet)
MR2332280

Zentralblatt MATH identifier
1114.62101

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62M30: Spatial processes 62M40: Random fields; image analysis

Keywords
Heavy-tailed random variables Poisson process random field stable random measure subsampling

Citation

McElroy, Tucker; Politis, Dimitris N. Stable marked point processes. Ann. Statist. 35 (2007), no. 1, 393--419. doi:10.1214/009053606000001163. https://projecteuclid.org/euclid.aos/1181100192


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