A central limit theorem for linear Kolmogorov's birth-growth models

Sung Nok CHIU

doi:10.1016/S0304-4149(96)00113-5

A central limit theorem for linear Kolmogorov's birth-growth models

Sung Nok CHIU^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

12 Citations (Scopus)

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Abstract

A Poisson process in space-time is used to generate a linear Kolmogorov's birth-growth model. Points start to form on [0,L] at time zero. Each newly formed point initiates two bidirectional moving frontiers of constant speed. New points continue to form on not-yet passed over parts of [0,L]. The whole interval will eventually be passed over by the moving frontiers. Let N_L be the total number of points formed. Quine and Robinson (1990) showed that if the Poisson process is homogeneous in space-time, the distribution of (N_L-E[N_L])/√var[N_L] converges weakly to the standard normal distribution. In this paper a simpler argument is presented to prove this asymptotic normality of N_L for a more general class of linear Kolmogorov's birth-growth models.

Original language	English
Pages (from-to)	97-106
Number of pages	10
Journal	Stochastic Processes and their Applications
Volume	66
Issue number	1
DOIs	https://doi.org/10.1016/S0304-4149(96)00113-5
Publication status	Published - 1 Feb 1997

Scopus Subject Areas

Statistics and Probability
Modelling and Simulation
Applied Mathematics

User-Defined Keywords

Central limit theorem
Coverage
Inhomogeneous Poisson process
Johnson-Mehl tessellation
Kolmogorov's birth-growth model

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10.1016/S0304-4149(96)00113-5

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title = "A central limit theorem for linear Kolmogorov's birth-growth models",

abstract = "A Poisson process in space-time is used to generate a linear Kolmogorov's birth-growth model. Points start to form on [0,L] at time zero. Each newly formed point initiates two bidirectional moving frontiers of constant speed. New points continue to form on not-yet passed over parts of [0,L]. The whole interval will eventually be passed over by the moving frontiers. Let NL be the total number of points formed. Quine and Robinson (1990) showed that if the Poisson process is homogeneous in space-time, the distribution of (NL-E[NL])/√var[NL] converges weakly to the standard normal distribution. In this paper a simpler argument is presented to prove this asymptotic normality of NL for a more general class of linear Kolmogorov's birth-growth models.",

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author = "CHIU, {Sung Nok}",

note = "Funding Information: I thank R. Cowan and D. Stoyan for the helpful discussions and valuable suggestions on the earlier drafts. I also thank the referee for the useful comments. This work was started during my stay in Freiberg, which was supported by a scholarship from DAAD.",

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N1 - Funding Information: I thank R. Cowan and D. Stoyan for the helpful discussions and valuable suggestions on the earlier drafts. I also thank the referee for the useful comments. This work was started during my stay in Freiberg, which was supported by a scholarship from DAAD.

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N2 - A Poisson process in space-time is used to generate a linear Kolmogorov's birth-growth model. Points start to form on [0,L] at time zero. Each newly formed point initiates two bidirectional moving frontiers of constant speed. New points continue to form on not-yet passed over parts of [0,L]. The whole interval will eventually be passed over by the moving frontiers. Let NL be the total number of points formed. Quine and Robinson (1990) showed that if the Poisson process is homogeneous in space-time, the distribution of (NL-E[NL])/√var[NL] converges weakly to the standard normal distribution. In this paper a simpler argument is presented to prove this asymptotic normality of NL for a more general class of linear Kolmogorov's birth-growth models.

AB - A Poisson process in space-time is used to generate a linear Kolmogorov's birth-growth model. Points start to form on [0,L] at time zero. Each newly formed point initiates two bidirectional moving frontiers of constant speed. New points continue to form on not-yet passed over parts of [0,L]. The whole interval will eventually be passed over by the moving frontiers. Let NL be the total number of points formed. Quine and Robinson (1990) showed that if the Poisson process is homogeneous in space-time, the distribution of (NL-E[NL])/√var[NL] converges weakly to the standard normal distribution. In this paper a simpler argument is presented to prove this asymptotic normality of NL for a more general class of linear Kolmogorov's birth-growth models.

KW - Central limit theorem

KW - Coverage

KW - Inhomogeneous Poisson process

KW - Johnson-Mehl tessellation

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A central limit theorem for linear Kolmogorov's birth-growth models

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