The determination of the age of a clone from characteristics of its geographical distribution

1981 ◽  
Vol 18 (03) ◽  
pp. 725-731
Author(s):  
Aid An Sudbury ◽  
Peter Clifford
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Geographical Distribution ◽  
Central Limit ◽  
Large Distance ◽  
Mutant Clone ◽  
New Type ◽  
The Individual ◽  
Ancient Mutation

Each point with integer coordinates in d dimensions is occupied by one individual. These individuals produce offspring at a Poisson rate 1, and these offspring migrate and displace other individuals. With probability u (the mutation rate) an offspring is of an entirely new type. A number of points N 0 will be occupied by the same type as the individual at the origin. It is shown that the distribution of N 0 arising from an ancient mutation does not differ greatly from the distribution of N 0 when the mutation is recent. However, the geographical spread is shown to be important, and a central limit theorem is proved for the age of the mutant clone given that a representative is present at a large distance from the origin.

The determination of the age of a clone from characteristics of its geographical distribution

1981 ◽  
Vol 18 (3) ◽  
pp. 725-731
Author(s):  
Aid An Sudbury ◽  
Peter Clifford
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Geographical Distribution ◽  
Central Limit ◽  
Large Distance ◽  
Mutant Clone ◽  
New Type ◽  
The Individual ◽  
Ancient Mutation

Each point with integer coordinates in d dimensions is occupied by one individual. These individuals produce offspring at a Poisson rate 1, and these offspring migrate and displace other individuals. With probability u (the mutation rate) an offspring is of an entirely new type. A number of points N0 will be occupied by the same type as the individual at the origin. It is shown that the distribution of N0 arising from an ancient mutation does not differ greatly from the distribution of N0 when the mutation is recent. However, the geographical spread is shown to be important, and a central limit theorem is proved for the age of the mutant clone given that a representative is present at a large distance from the origin.

scholarly journals Central limit theorem for a class of random measures associated with germ-grain models

1999 ◽  
Vol 31 (02) ◽  
pp. 283-314 ◽  
Author(s):  
Lothar Heinrich ◽  
Ilya S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Boolean Model ◽  
Random Sets ◽  
Model Parameters ◽  
Random Measures ◽  
Intrinsic Volumes ◽  
The Individual ◽  
Individual Grains

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝ d generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

scholarly journals Central limit theorem for a class of random measures associated with germ-grain models

1999 ◽  
Vol 31 (2) ◽  
pp. 283-314 ◽  
Author(s):  
Lothar Heinrich ◽  
Ilya S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Boolean Model ◽  
Random Sets ◽  
Model Parameters ◽  
Random Measures ◽  
Intrinsic Volumes ◽  
The Individual ◽  
Individual Grains

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝd generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

Determination of the Minimum Sampling Size for Performance Verification of Security Gate

2010 ◽  
Vol 40-41 ◽  
pp. 913-916
Author(s):  
Juan Chen ◽  
Chun Wen Song ◽  
Wei Wu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Aviation Safety ◽  
Safety Verification ◽  
Performance Verification ◽  
Proper Sampling

This Security Gate is thought to be of great importance to aviation safety. Verification need to be done to make sure of quality of security gate. In order to do verification well, specific sampling size need to be determined first. So how to select proper sampling size is supposed to be significant. According to central limit theorem this problem is solved in this paper.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

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