Stable marked point processes

TL;DR

This paper investigates the asymptotic behavior of sample means and variances in marked point processes generated by Poisson random measures, extending finite variance results to heavy-tailed distributions and exploring subsampling for confidence intervals.

Contribution

It introduces new asymptotic results for sample statistics in heavy-tailed marked point processes and develops subsampling methods for confidence interval estimation.

Findings

Extended finite variance results to heavy-tailed settings.

Derived joint asymptotic behavior of sample mean and variance.

Proposed subsampling approach for confidence intervals under unknown tail heaviness.

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 $\{(t_i,X(t_i))\}$, where X(t) is a stationary random field and the points t_i are generated from an independent Poisson random measure $\mathbb{N}$ on $\mathbb{R}^d$. 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.