Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: plasm-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: eess
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The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph
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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: cmp-lg