Asymptotic behavior of some strongly critical decomposable 3-type Galton–Watson processes with immigration

Abstract We study the asymptotic behavior of a critical decomposable 3-type Galton–Watson process with immigration when its offspring mean matrix is triangular with diagonal entries all 1. It is proved that, under second-order or fourth-order moment assumptions on the offspring and immigration distributions, a sequence of appropriately scaled random step processes formed from such a Galton–Watson process converges weakly. The limit process can be described with the use of independent squared Bessel processes left parenthesis script upper X Subscript t comma 1 Baseline right parenthesis Subscript t greater than or equals 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script" class="MJX-tex-caligraphic">X</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> $({\mathcal X}_{t,1})_{t\geq0}$ , left parenthesis script upper X Subscript t comma 2 Baseline right parenthesis Subscript t greater than or equals 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script" class="MJX-tex-caligraphic">X</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> $({\mathcal X}_{t,2})_{t\geq0}$ , and left parenthesis script upper X Subscript t comma 3 Baseline right parenthesis Subscript t greater than or equals 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script" class="MJX-tex-caligraphic">X</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> $({\mathcal X}_{t,3})_{t\geq0}$ , the linear combinations of the integral processes of left parenthesis script upper X Subscript t comma 1 Baseline right parenthesis Subscript t greater than or equals 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script" class="MJX-tex-caligraphic">X</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> $({\mathcal X}_{t,1})_{t\geq0}$ and left parenthesis script upper X Subscript t comma 2 Baseline right parenthesis Subscript t greater than or equals 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mro