In this somewhat didactic note we give a detailed alternative proof of the known
result of 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 (not necessarily starting from 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 Ispány and Pap (2010). This
technique was already used by Ispány (2008), who 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 of
Ispány (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 nonzero initial
values with the technique used by Ispány (2008), and further, we simplify
some of the arguments in the proof of Theorem 2.1 of Ispány (2008) as
well.
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