Some asymptotic results for the branching process with immigration
Let \(\{X_ n\}\) be a branching process with immigration \[ X_ n=\sum^{X_{n-1}}_{j=1}Y_{n,j}+I_ n,\quad T_ n=\sum^{n}_{i=1}(1+X_{i-1})^{-1};\quad \lambda =E I_ 1,\quad m=E Y_{1,1},\quad \sigma^ 2=E(Y_{1,1}-m)^ 2. \] The authors give results on: (i) the convergence of \(X_{[nt]}/n\) in \(D^+[0,\infty];\) (ii) the convergence of \(T_ n/\log n\) if \(2\lambda /\sigma^ 2>1, m=1;\) (iii) the convergence of \(n^{\tau -1}T_ n\) if \(2\lambda /\sigma^ 2<1, m=1;\) (iv) the convergence in \(R^{\infty}\) with the metric \[ d(x,y)=\sum 2^{-i}| x_ i-y_ i| /(1+| x_ i-y_ i|)\quad if\quad m>1. \]
scientific article; zbMATH DE number 4102249
scientific article