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December, 1975 Poisson Convergence and Family Trees
A. R. Moncayo
Ann. Probab. 3(6): 1059-1061 (December, 1975). DOI: 10.1214/aop/1176996235


Cells of certain variety live a random length of time and then split into two new cells. Let $t_1 < t_2 < t_3 < \cdots$, be an increasing sequence of positive numbers such that any given cell has probability $\lambda/n$ with $\lambda > 0$, that its life span be at least $t_n$ units of time. Starting with one cell, the $n$th generation will have $2^n$ cells and for each one we count the number of its ancestors and itself whose life span was at least $t_n$ units of time. These numbers determine an empirical distribution (the $n$th empirical distribution). It is shown that for almost all cell cultives (starting each time with one cell) the sequence of these empirical distributions converges to the Poisson distribution with parameter $\lambda$.


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A. R. Moncayo. "Poisson Convergence and Family Trees." Ann. Probab. 3 (6) 1059 - 1061, December, 1975.


Published: December, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0343.60055
MathSciNet: MR386040
Digital Object Identifier: 10.1214/aop/1176996235

Primary: 60G99
Secondary: 60F05 , 60J80

Keywords: Random empirical distribution

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 6 • December, 1975
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