50 research outputs found
The Walking Dead: Should Awards that Have Been Annulled at the Seat Nevertheless be Enforced by Courts in Other Jurisdictions?
When Justice Delayed is Justice Denied: Advantages and Disadvantages of Emergency Arbitration in Comparison with Interim Relief
A note on asymptotic behavior of critical Galton-Watson processes with immigration
In this somewhat didactic note we give a detailed alternative proof of the
known result due to Wei and Winnicki (1989) which states that under second
order moment assumptions on the offspring and immigration distributions the
sequence of appropriately scaled random step functions formed from a critical
Galton-Watson process with immigration (starting from not necessarily zero)
converges weakly towards a squared Bessel process. The proof of Wei and
Winnicki (1989) is based on infinitesimal generators, while we use limit
theorems for random step processes towards a diffusion process due to Isp\'any
and Pap (2010). This technique was already used in Isp\'any (2008), where he
proved functional limit theorems for a sequence of some appropriately
normalized nearly critical Galton-Watson processes with immigration starting
from zero, where the offspring means tend to its critical value 1. As a special
case of Theorem 2.1 in Isp\'any (2008) one can get back the result of Wei and
Winnicki (1989) in the case of zero initial value. In the present note we
handle non-zero initial values with the technique used in Isp\'any (2008), and
further, we simplify some of the arguments in the proof of Theorem 2.1 in
Isp\'any (2008) as well.Comment: 20 page
Asymptotic behaviour of critical decomposable 2-type Galton-Watson processes with immigration
In this paper the asymptotic behaviour of a critical 2-type Galton-Watson
process with immigration is described when its offspring mean matrix is
reducible, in other words, when the process is decomposable. It is proved that,
under second or fourth order moment assumptions on the offspring and
immigration distributions, a sequence of appropriately scaled random step
processes formed from a critical decomposable 2-type Galton-Watson process with
immigration converges weakly. The limit process can be described using one or
two independent squared Bessel processes and possibly the unique stationary
distribution of an appropriate single-type subcritical Galton-Watson process
with immigration. Our results complete and extend the results of Foster and Ney
(1978) for some strongly critical decomposable 2-type Galton-Watson processes
with immigration.Comment: 51 page