Poisson Convergence and Family Trees

Abstract

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 $\lambda$.