  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 the sequence of appropriately scaled random step
functions formed from a critical Galton-Watson process with immigration
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 martingale differences 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 by simplifying some of the arguments in Isp\'any
(2008) as well.
