Abstract
Abstract A properly scaled critical Galton–Watson process converges to a continuous state critical branching process ξ ( ⋅ ) \xi(\,{\cdot}\,) 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 ( ∫ 0 y ξ ( y - u ) d u γ \int_{0}^{y}\xi(y-u)\,du^{\gamma} , y ≥ 0 y\geq 0 ) with a pertinent γ ≥ 0 \gamma\geq 0 .
Highlights
One of the basic stochastic population models of a self-reproducing system is built upon the following two assumptions: (A) different individuals live independently from each other according to the same individual life law described in (B); (B) an individual dies at age one and at the moment of death gives birth to a random number N of offspring
We study {Z(t), t ≥ 0}, a Galton–Watson process with overlapping generations, or GWOprocess for short, where Z(t) is the number of individuals alive at time t in a reproduction system satisfying the following two assumptions: (A*) different individuals live independently from each other according to the same individual life law described in (B*); (B*) an individual lives L units of time and gives N births at random ages τ1, . . . , τN, satisfying
The aim of this chapter is to establish an fdd-convergence result for the vector (X1( ⋅ ), . . . , Xq( ⋅ )) composed of the population counts corresponding to different individual scores χ1( ⋅ ), . . . , χq( ⋅ ), which may depend on each other
Summary
One of the basic stochastic population models of a self-reproducing system is built upon the following two assumptions: (A) different individuals live independently from each other according to the same individual life law described in (B); (B) an individual dies at age one and at the moment of death gives birth to a random number N of offspring. We study {Z(t), t ≥ 0}, a Galton–Watson process with overlapping generations, or GWOprocess for short, where Z(t) is the number of individuals alive at time t in a reproduction system satisfying the following two assumptions: (A*) different individuals live independently from each other according to the same individual life law described in (B*); (B*) an individual lives L units of time and gives N births at random ages τ1, . The following three statements are straightforward corollaries of Theorems 1, 2, and 3 respectively In these theorems, it is always assumed that the GWO-process stems from a large number Z0 = n of progenitors born at time zero. (5) In different formulas, the symbols C, C1, C2, c, c1, c2 represent different positive constants
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