A low-rank technique for computing the quasi stationary distribution of Galton-Watson processes

Data Seminario: 
06-Dec-2018
Ora Inizio: 
10:00
Ora Fine: 
11:00
Stefano Massei
EPFL

Place: Sala Seminari Ovest, Dipartimento di Informatica. 

 

Branching processes describe the dynamics of a population of individuals which reproduce and die independently, according to some specific probability distributions. More precisely, we assume that any individual has a unit lifetime, at the end of which it might give birth to one or more offsprings simultaneously.

This is encoded into the probability generating function P(z):=\sum p_j z^j where p_j is the probability of generating j individuals. These kind of processes are known in the literature as Galton-Watson processes. We consider populations that are certain to become extinct, yet appear to be stationary over any reasonable time scale.

More precisely, we are interested in characterizing the quasi-stationary distribution of the process, i.e., the asymptotic distribution of the population size, conditional on its survival.
Yaglom proved that if m:=P'(1)<1 then the quasi stationary distribution exists and its probability generating function G(z):=\sum g_j z^j solves the Schroeder functional equation

G(P(z))= mG(z)+1-m,  z in [0,1].           (*)

We study the link between the regularity of P(z) and that of G(z) and we propose a strategy for  solving (*) in the case where P(z) and G(z) are analytic on a disc  of radius r>1.
We see that the discretization of (*) leads to a numerical method that is capable to find arbitrary accurate approximations of the coefficients of G(z).

Moreover, we point out the (numerical) low-rank structure that appears in the discretized problem, and we show how to exploit it in the proposed procedure.