Critical Galton–Watson Processes with Overlapping Generations

Abstract A properly scaled critical Galton–Watson process converges to a continuous state critical branching process <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>ξ</mi> <mo>⁢</mo> <mrow> <mo stretchy="false">(</mo> <mo lspace="4.2pt" rspace="4.2pt">⋅</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math><math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>ξ</mi> <mo>⁢</mo> <mrow> <mo stretchy="false">(</mo> <mo lspace="4.2pt" rspace="4.2pt">⋅</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math> as the number of initial individuals tends to infinity. We extend this classical result by allowing for overlapping generations and considering a wide class of population counts. The main result of the paper establishes a convergence of the finite-dimensional distributions for a scaled vector of multiple population counts. The set of the limiting distributions is conveniently represented in terms of integrals ( <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msubsup> <mo largeop="true" symmetric="true">∫</mo> <mn>0</mn> <mi>y</mi> </msubsup> <mrow> <mi>ξ</mi> <mo>⁢</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <mi>y</mi> <mo>-</mo> <mi>u</mi> </mrow> <mo rspace="4.2pt" stretchy="false">)</mo> </mrow> <mo>⁢</mo> <mrow> <mo mathvariant="italic" rspace="0pt">d</mo> <msup> <mi>u</mi> <mi>γ</mi> </msup> </mrow> </mrow> </mrow> </math><math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msubsup> <mo largeop="true" symmetric="true">∫</mo> <mn>0</mn> <mi>y</mi> </msubsup> <mrow> <mi>ξ</mi> <mo>⁢</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <mi>y</mi> <mo>-</mo> <mi>u</mi> </mrow> <mo rspace="4.2pt" stretchy="false">)</mo> </mrow> <mo>⁢</mo> <mrow> <mo mathvariant="italic" rspace="0pt">d</mo> <msup> <mi>u</mi> <mi>γ</mi> </msup> </mrow> </mrow> </mrow> </math>, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math><math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math> ) with a pertinent <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>γ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math><math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>γ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math> .