Central limit theory for the number of seeds in a growth model in ℝd with inhomogeneous Poisson arrivals

Sung Nok CHIU*, M. P. Quine

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

31 Citations (Scopus)

Abstract

A Poisson point process ψ in d-dimensional Euclidean space and in time is used to generate a birth-growth model: seeds are born randomly at locations xi in ℝd at times ti ∈ [0, ∞). Once a seed is born, it begins to create a cell by growing radially in all directions with speed v > 0. Points of ψ contained in such cells are discarded, that is, thinned. We study the asymptotic distribution of the number of seeds in a region, as the volume of the region tends to infinity. When d = 1, we establish conditions under which the evolution over time of the number of seeds in a region is approximated by a Wiener process. When d ≥ 1, we give conditions for asymptotic normality. Rates of convergence are given in all cases.

Original languageEnglish
Pages (from-to)802-814
Number of pages13
JournalAnnals of Applied Probability
Volume7
Issue number3
Publication statusPublished - Aug 1997

Scopus Subject Areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

User-Defined Keywords

  • Birth-growth
  • Brownian motion
  • Central limit theorem
  • Inhomogeneous Poisson process
  • Rate of convergence

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