[Probabilidad-Estadistica-Seminario] Seminario de Probabilidad y Estadística -- Viernes 21 de abril

[Andrés Sosa]

Hola

Este* viernes 21 de abril a las 10:00 horas* en el salón de seminarios del
Centro de Matemática hablará *Matthieu Jonckheere* (Instituto de Cálculo --
Facultad de Ciencias Exactas -- UBA) en el seminario de Probabilidad y
Estadística.

El título de la charla es: *On the Kesten-Stigum theorem in $L^2$ beyond
$R$-positivity.*

Es un trabajo en conjunto con S. Saglietti.
La presentación será en español.

Saludos.
Andrés







*Abstract:We study supercritical branching processes where each particle
branches according to an independent Poisson process and evolves describing
a Markovian motion which may possess absorbing states  or be transient.
Under natural assumptions on this underlying motion, we first show using
only probablistic tools that there is convergence in $L^2$ for the
empirical measure (when normalized by the mean number of particles) if and
only if a naturally-defined variance of the process is finite.  This
significantly generalizes the state of the art which was mainly restricted
to specific $R$-positive processes.We then show that, whenever this natural
variance is finite, the true empirical measure (i.e. normalized by the real
number of particles) also converges (in probability) if and only if the
branching process survives locally. Finally, building on previous results
we prove that, in the case of $R$-positive processes, these convergences
also hold in the almost sure sense provided that the $h$-transform of the
driving Markovian motion admits a suitable Lyapunov functional, a condition
which in many examples is fairly easy to verify.These results yield a
simple procedure to simulate (quasi-)stationary distributions, whose closed
forms are in general unknown.A strong law of large numbers for the
empirical measure in this last example was announced by Kesten, although a
proof of it has remained undisclosed so far. Our results are able to give a
partial proof to Kesten's claim, by yielding a weak law of large numbers
whenever the aforementioned natural variance of the process is finite.*
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