Limit behaviour for a subcritical bisexual Galton-Watson branching process with immigration
The bisexual branching process \(Z^*_n\) with immigration is specified by the formulas \[ Z^*_0=N, \quad (F^*_{n+1}, M^*_{n+1})=\sum_{i=1}^{Z^*_n}(f_{ni},m_{ni}) +(F^I_{n+1}, M^I_{n+1}), \] \[ Z^*_{n+1}=L(F^*_{n+1}, M^*_{n+1}), \quad n=0,1,\ldots . \] Here \(\{(f_{ni},m_{ni})\), \(i=1,2,\ldots\), \(n=0,1,\ldots \}\) is a sequence of iid non-negative integer-valued random variables interpreted as the number of females (males) \(f_{ni}\) \((m_{ni})\) originated by the \(i\)th maiting unit in the \(n\)th generation, the components of the iid vectors \((F^I_{n}, M^I_{n})\), \(n=1,\ldots \), represent the number of immigrant females and males, respectively, in the \(n\)th generation, and \(L(x,y)\) is a mating function. Assuming that \(L(x,y)\) is a superadditive function and that \[ \limsup_{k\to\infty}k^{-1}L \Biggl( \sum_{i=1}^{k}(f_{ni},m_{ni}) +(F^I_{n+1}, M^I_{n+1})\Biggr)<1, \] the authors show convergence of \(Z^*_n\) as \(n\to\infty\) to a finite positive and non-degenerate random variable.
scientific article; zbMATH DE number 1506805
scientific article