Research

Colloquia — Summer 2004

Monday, May 10, 2004

Abstract

A multitype branching process \(\left(Z_n\right)_{n\ge 0}\) is a Markov chain on \(\mathbb{N}^k\) governed by \(k\) probability distributions on \(\mathbb{N}^k\) called fertility laws with respective generating functions \(f_1\left(s_1,\dotsc,s_k\right),\dotsc,f_k\), such that $$ \mathbb{E}\left(s_1^{\left(Z_{n+1}\right)_1}\dotsm s_k^{\left(Z_{n+1}\right)_k}\mid Z_n\right) =f_1(s)^{\left(Z_n\right)_1}\dotsm f_k(s)^{\left(Z_n\right)_k}. $$ For \(k=1\) explicit calculations about this process are trivial when the fertility is given by \(f(s)=\frac{as+b}{cs+d}\). The lecture will extend this to \(k>1\) for fractional linear transformations of \(\mathbb{R}^k\), which can be illustrated for \(k=2\) as $$ f_1\left(s_1,s_2\right)=\frac{a_{11}s_1+a_{12}s_2+b_1}{c_1s_1+c_2s_2+d},\quad f_2\left(s_1,s_2\right)=\frac{a_{21}s_1+a_{22}s_2+b_2}{c_1s_1+c_2s_2+d}. $$

Surprisingly enough, this simple case has not been studied. We denote by \(\rho\) the largest eigenvalue of the matrix \(M\) by the means of the fertility laws. We shall compute here:

1. The extinction probability \(\lim\limits_{n\to\infty}\mathrm{Pr}\left(Z_n=0\right)\).
2. The distribution of the total progeny \(\sum\limits_{n=0}^\infty Z_n\) when \(\rho\le 1\).
3. The limit distribution of \(\rho^{-n}Z_n\) when \(\rho > 1\).
4. The limit distribution of \(Z_n\mid Z_n\ne 0\) when \(\rho < 1\).
5. Stationary measures.