# Research

## Colloquia — Summer 2004

### Monday, May 10, 2004

Title: Fractional linear multitype branching process
Speaker: Gérard Letac, Université Paul Sabatier
Toulouse, France
Time: 3:00pm‐4:00pm
Place: PHY 108

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 $$\rho < 1$$.